\label{l1}Let $\Omega$ be a bounded domain in $\mathbb{R}^n$, $f_{\pm} :
  \bar{\Omega} \rightarrow \mathbb{R}^n$ be $C^2$-maps such that $f_+
  |_{\partial \Omega}   = f_- |_{\partial \Omega}  $. then
  \begin{equation}
    \int_{\Omega} \det f_+' (x) \mathrm{d} x = \int_{\Omega} \det f_-' (x) \mathrm{d}
    x \text{.} \label{e1}
  \end{equation}

\begin{theorem}
  \label{l1}Let $\Omega$ be a bounded domain in $\mathbb{R}^n$, $f_{\pm} :
  \bar{\Omega} \rightarrow \mathbb{R}^n$ be $C^2$-maps such that $f_+
  |_{\partial \Omega}   = f_- |_{\partial \Omega}  $. then
  \begin{equation}
    \int_{\Omega} \det f_+' (x) \mathd x = \int_{\Omega} \det f_-' (x) \mathd
    x \text{.} \label{e1}
  \end{equation}
\end{theorem}

\begin{equation}
  \int_B \det R' (x) \mathd x = \int_B \mathd x = \mathrm{Vol} (B) > 0 \text{,}
  \label{e2}
\end{equation}

In a recent paper {\cite{MR4725395}}, to respond inquiry from some readers
about the solution of Exercise 1.2.1 in his book {\cite{MR2435520}}, Krylov
gives a proof of the following theorem, see \citet[Lemma 1]{MR4725395}, where $f_\pm$ only need to be $C^1$.

\begin{abstract}
 It is known that the integral of the Jacobian determinant of a smooth map $f
 : \bar{\Omega} \rightarrow \mathbb{R}^n$ depends only on $f |_{\partial
 \Omega} $ and this result leads to an analytic proof of the
 Brouwer fixed point theorem. In this note we provide two new proofs of this
 result, one by classical analysis and one by differential forms and Stokes
 formula.
\end{abstract}

In a recent paper {\cite{MR4725395}}, to respond inquiry from some readers
about the solution of Exercise 1.2.1 in his book {\cite{MR2435520}}, Krylov
gives a proof of the following theorem, see \citet[Lemma 1]{MR4725395}, where $f_\pm$ only need to be $C^1$.

Krylov's proof of Theorem \ref{l1} in {\cite{MR4725395}} is based on the
observation: for small $t$, $f_{\pm}^t = \mathrm{id} + t f_{\pm}$ are
diffeomorphisms and $f^t_+ (\Omega) = f_-^t (\Omega)$ because $f_+ |_{\partial
\Omega} = f_- |_{\partial \Omega} $, thus by the changing
variable formula
\[ \int_{\Omega} \det (I + t f_+' (x)) \mathd x = \mathrm{Vol} (f_+^t (\Omega))
 = \mathrm{Vol} (f_-^t (\Omega)) = \int_{\Omega} \det (I + t f_-' (x)) \mathd
 x \text{.} \]
Since the two sides are polynomials in $t$, comparing the coefficients of
$t^n$ gives (\ref{e1}). To make the argument rigorous some issues including
why $f_{\pm}^t (\partial \Omega) = \partial f_{\pm}^t (\Omega)$ and why
$\partial f_+^t (\Omega) = \partial f_-^t (\Omega)$ implies $f_+^t (\Omega) =
f_-^t (\Omega)$ need to be handled. These are clarified via a careful point
set analysis in a long remark following the proof. One of the advantages of
this argument is that the maps $f_{\pm}$ only need to be $C^1$.

Theorem \ref{l1} also appears in {\cite{MR2597943,MR1046451}}. In
{\cite{MR1046451}}, Kulpa defines
\[ I (h^1, \ldots, h^n) = \int_{\Omega} \det h' (x) \mathd x \]
for smooth $h : \bar{\Omega} \rightarrow \mathbb{R}^n$ and manages to show
\begin{equation}
 I (f^1_+, f^2_+, \ldots, f_+^n) = I (f^1_-, f^2_+, \ldots, f_+^n) \label{e3}
\end{equation}
using Fubini theorem and projections $\Pi_i : \mathbb{R}^n \rightarrow
\mathbb{R}^{n - 1}$ given by deleting the $i$-th component. Then (\ref{e1})
follows by applying (\ref{e3}) successively to replace the remaining columns
of $\det f_+' (x)$ by those of $\det f_-' (x)$. While in {\citet[pp.\
464]{MR2597943}}, (\ref{e1}) is proved using the facts that $L (P) = \det P$
is a null Lagrangian and the energy $\int_{\Omega} L (w' (x), w (x), x) \mathd
x$ of null Lagrangian $L : \mathbb{M}^{m \times n} \times \mathbb{R}^m \times
\bar{\Omega} \rightarrow \mathbb{R}$ depends only on boundary conditions.

Let $f^i_{\pm}$ be the $i$-th component of $f_{\pm}$ and $\Phi_{\pm} =
(f_{\pm}^2, \ldots, f^n_{\pm})$. Then $\Phi_+ = \Phi_-$ on $\partial \Omega$.
Let
\[ A_{\pm} = \left( \frac{\partial (f_{\pm}^2, \ldots, f_{\pm}^n)}{\partial
 (x^2, \ldots, x^n)}, - \frac{\partial (f_{\pm}^2, f_{\pm}^3, \ldots,
 f_{\pm}^n)}{\partial (x^1, x^3, \ldots, x^n)}, \ldots, (- 1)^{n + 1}
 \frac{\partial (f_{\pm}^2, \ldots, f_{\pm}^n)}{\partial (x^1, \ldots, x^{n
 - 1})} \right) \text{,} \]
then the $i $-th component of $A_{\pm}$ is just the cofactor of
$\partial_{x^i} f_{\pm}^1$ in $\det f_{\pm}' (x)$. By Jacobi identity (see
e.g.\ {\citet[Eq.\ (5)]{MR610487}} or {\citet[Eq.\ (0.3)]{MR4701455}}), we have
$\mathrm{div} A_{\pm} = 0$. Thus
\begin{equation}
 \mathrm{div} (f_{\pm}^1 A_{\pm}) = \nabla f_{\pm}^1 \cdummy A_{\pm} +
 f_{\pm}^1 \mathrm{div} A_{\pm} = \nabla f_{\pm}^1 \cdummy A_{\pm} = \det
 f_{\pm}' (x) \text{.} \label{d}
\end{equation}
For $a \in \partial \Omega$, let $\eta : (u^1, \ldots, u^{n - 1}) \mapsto x$
be a local parametrization of $\partial \Omega$ at $a = \eta (\alpha)$ such
that
\[ N(a) = \left( \frac{\partial (x^2, \ldots, x^n)}{\partial (u^1, \ldots, u^{n -
 1})}, - \frac{\partial (x^1, x^3, \ldots, x^n)}{\partial (u^1, \ldots, u^{n
 - 1})}, \ldots, (- 1)^{n + 1} \frac{\partial (x^1, \ldots, x^{n -
 1})}{\partial (u^1, \ldots, u^{n - 1})} \right)_{\alpha} \]
is an outward normal vector of $\partial \Omega$ at $a$. Applying the chain
rule to the composition $y_{\pm} = \Phi_{\pm} (\eta (u))$, we see that
\[ \left( \begin{array}{ccc}
 \dfrac{\partial y_{\pm}^2}{\partial u^1} & \cdots & \dfrac{\partial
 y_{\pm}^2}{\partial u^{n - 1}}\\
 & & \\
 \dfrac{\partial y_{\pm}^n}{\partial u^1} & \cdots & \dfrac{\partial
 y_{\pm}^n}{\partial u^{n - 1}}
 \end{array} \right)_{\alpha} = \left( \begin{array}{ccc}
 \dfrac{\partial y_{\pm}^2}{\partial x^1} & \cdots & \dfrac{\partial
 y_{\pm}^2}{\partial x^n}\\
 & & \\
 \dfrac{\partial y_{\pm}^n}{\partial x^1} & \cdots & \dfrac{\partial
 y_{\pm}^n}{\partial x^n}
 \end{array} \right)_a \left( \begin{array}{ccc}
 \dfrac{\partial x^1}{\partial u^1} & \cdots & \dfrac{\partial
 x^1}{\partial u^{n - 1}}\\
 & & \\
 \dfrac{\partial x^n}{\partial u^1} & \cdots & \dfrac{\partial
 x^n}{\partial u^{n - 1}}
 \end{array} \right)_{\alpha} \text{,} \]
where the matrix on the left is the Jacobian matrix $(\Phi_{\pm} \circ \eta)'
(\alpha)$.

Because $\Phi_+ = \Phi_-$ on $\partial \Omega$ and $\eta (u) \in \partial
\Omega$ for $u$ near $\alpha$, we have $\Phi_+ \circ \eta = \Phi_- \circ \eta$
near $\alpha$. Using the Cauchy-Binet formula we deduce
\begin{align*}
 (A_+ \cdummy N) (a) & = \sum_{i = 1}^n \left. \frac{\partial (f_+^2,
 \ldots, f_+^n)}{\partial (x^1, \ldots, \hat{x}^i, \ldots, x^n)} \right|_a
 \cdummy \left. \frac{\partial (x^1, \ldots, \hat{x}^i, \ldots,
 x^n)}{\partial (u^1, \ldots, u^{n - 1})} \right|_{\alpha}\\
 & = \det (\Phi_+ \circ \eta)' (\alpha) = \det (\Phi_- \circ \eta)'
 (\alpha) = (A_- \cdummy N) (a) \text{.}
\end{align*}
Therefore
\begin{equation}
 A_+ \cdummy \nu = \frac{A_+ \cdummy N}{| N |} = \frac{A_- \cdummy N}{| N |}
 = A_- \cdummy \nu \label{e5}
\end{equation}
on $\partial \Omega$, where $\nu = N / | N |$ is outward unit normal vector
field on $\partial \Omega$.

Since $f^1_+ = f^1_-$ on $\partial \Omega$, using the divergence theorem we
deduce from (\ref{d}) and (\ref{e5})
\begin{align*}
 \int_{\Omega} \det f_+' (x) \mathd x & = \int_{\Omega} \mathrm{div} (f_+^1
 A_+) \mathd x = \int_{\partial \Omega} f_+^1 A_+ \cdummy \nu \mathd \sigma\\
 & = \int_{\partial \Omega} f_-^1 A_- \cdummy \nu \mathd \sigma =
 \int_{\Omega} \det f_-' (x) \mathd x \text{.}
\end{align*}
\begin{example}
 Let $A$ be an $n \times n$ matrix, $\varphi \in C_0^{\infty} (\Omega,
 \mathbb{R}^n)$. Set $f_+ (x) = A x + \varphi (x)$, $f_- (x) = A x$, we get
 the formula stated in {\citet[pp. 21]{MR3887613}}:
 \[ \int_{\Omega} \det (A + \varphi' (x)) \mathd x = \det (A) \cdummy m
 (\Omega) \text{.} \]
\end{example}

To conclude this note, we present another proof using differential forms. Let
$f^i$ be the components of $f:\bar{\Omega}\to\mathbb{R}^n$, then it is well known that
\begin{align*}
 \det f' (x) \mathd x^1 \wedge \cdots \wedge \mathd x^n & = \mathd f^1
 \wedge \mathd f^2 \wedge \cdots \wedge \mathd f^n \text{.}\\
 & = \mathd (f^1 \mathd f^2 \wedge \cdots \wedge \mathd f^n) \text{,}
\end{align*}
where in the second equality we used $\mathd^2 = 0$. Recall the Stokes formula
\[ \int_{\Omega} \mathd \omega = \int_{\partial \Omega} i^{\ast} \omega
 \text{,} \]
where $\omega$ is an $(n - 1)$-form on $\Omega$ and $i^{\ast}$ is the pull
back induced by the embedding $i : \partial \Omega \rightarrow \Omega$.
Because $i^{\ast} \circ \mathd = \mathd \circ i^{\ast}$, we conclude
\begin{align*}
 \int_{\Omega} \det f' (x) \mathd x & = \int_{\Omega} \det f' (x) \mathd
 x^1 \wedge \cdots \wedge \mathd x^n\\
 & = \int_{\Omega} \mathd (f^1 \mathd f^2 \wedge \cdots \wedge \mathd f^n)
 = \int_{\partial \Omega} i^{\ast} (f^1 \mathd f^2 \wedge \cdots \wedge
 \mathd f^n)\\
 & = \int_{\partial \Omega} (f^1 \circ i) \mathd (f^2 \circ i) \wedge
 \cdots \wedge \mathd (f^n \circ i) \text{.}
\end{align*}
Clearly the right hand side depends only on $f \circ i = f |_{\partial \Omega}
 $.

Using this theorem, the \emph{no-retraction theorem} follows immediately, see \citet[Corollary
1]{MR4725395}. Let $B$ be the closed unit ball in
$\mathbb{R}^n$. If there was a smooth retraction $R : B \rightarrow \partial
B$, since $R |_{\partial B}   = \mathrm{id} |_{\partial B}  $,
Theorem \ref{l1} with $f_+ = R$, $f_- = \mathrm{id}$ yields a contradiction
\begin{equation}
  \int_B \det R' (x) \mathrm{d} x = \int_B \mathrm{d} x = \mathrm{Vol} (B) > 0 \text{,}
  \label{e2}
\end{equation}
because $\det R' (x) = 0$, a consequence of $R (B) \subset \partial B$. That the two
integrals in (\ref{e2}) are equal also follows from a version of the changing
variable formula given in {\citet[Theorem 3.1]{MR3711061}}: \emph{For smooth closed
bounded domain $D$ in $\mathbb{R}^n$ and smooth map $\varphi : B \rightarrow
D$, if $\varphi : \partial B \rightarrow \partial D$ is a diffeomorphism, then
for continuous $f : D \rightarrow \mathbb{R}$ there holds}
\[ \int_D f (y) \mathrm{d} y = \pm \int_B f (\varphi (x)) \det \varphi' (x) \mathrm{d}
   x \text{.} \]
The first equality in (\ref{e2}) follows by letting $f = 1$, $D = B$ and
$\varphi = R$ in this formula. It is well known that the no-retraction theorem
is equivalent to the famous Brouwer fixed point theorem.

Krylov's proof of Theorem \ref{l1} in {\cite{MR4725395}} is based on the
observation: for small $t$, $f_{\pm}^t = \mathrm{id} + t f_{\pm}$ are
diffeomorphisms and $f^t_+ (\Omega) = f_-^t (\Omega)$ because $f_+ |_{\partial
\Omega}   = f_- |_{\partial \Omega}  $, thus by the changing
variable formula
\[ \int_{\Omega} \det (I + t f_+' (x)) \mathrm{d} x = \mathrm{Vol} (f_+^t (\Omega))
   = \mathrm{Vol} (f_-^t (\Omega)) = \int_{\Omega} \det (I + t f_-' (x)) \mathrm{d}
   x \text{.} \]
Since the two sides are polynomials in $t$, comparing the coefficients of
$t^n$ gives (\ref{e1}). To make the argument rigorous some issues including
why $f_{\pm}^t (\partial \Omega) = \partial f_{\pm}^t (\Omega)$ and why
$\partial f_+^t (\Omega) = \partial f_-^t (\Omega)$ implies $f_+^t (\Omega) =
f_-^t (\Omega)$ need to be handled. These are clarified via a careful point
set analysis in a long remark following the proof. One of the advantages of
this argument is that the maps $f_{\pm}$ only need to be $C^1$.

Theorem \ref{l1} also appears in {\cite{MR2597943,MR1046451}}. In
{\cite{MR1046451}}, Kulpa defines
\[ I (h^1, \ldots, h^n) = \int_{\Omega} \det h' (x) \mathrm{d} x \]
for smooth $h : \bar{\Omega} \rightarrow \mathbb{R}^n$ and manages to show
\begin{equation}
  I (f^1_+, f^2_+, \ldots, f_+^n) = I (f^1_-, f^2_+, \ldots, f_+^n) \label{e3}
\end{equation}
using Fubini theorem and projections $\Pi_i : \mathbb{R}^n \rightarrow
\mathbb{R}^{n - 1}$ given by deleting the $i$-th component. Then (\ref{e1})
follows by applying (\ref{e3}) successively to replace the remaining columns
of $\det f_+' (x)$ by those of $\det f_-' (x)$. While in {\citet[pp.\
464]{MR2597943}}, (\ref{e1}) is proved using the facts that $L (P) = \det P$
is a null Lagrangian and the energy $\int_{\Omega} L (w' (x), w (x), x) \mathrm{d}
x$ of null Lagrangian $L : \mathbb{M}^{m \times n} \times \mathbb{R}^m \times
\bar{\Omega} \rightarrow \mathbb{R}$ depends only on boundary conditions.

The purpose of this note is to present another two proofs of Theorem \ref{l1}.
The first one is performed by playing classical analysis, which is written for
readers not familiar with differential forms and in our opinion is very
elementary and transparent. The second one is quite short, which depends on
differential form and Stokes formula on manifolds. As demonstrated in the
paragraph after Theorem \ref{l1}, our note is useful for understanding the
Brouwer fixed point theorem from the analytic point of view. In addition to
{\cite{MR2597943,MR4725395,MR1046451,MR3711061}} mentioned above, other
analytic proofs of the Brouwer fixed point theorem can be found in \cite[pp.\ 467--470]{MR117523} and
{\cite{MR610487,MR1699248,MR505523,MR600910}}.

Krylov's proof of Theorem~\ref{l1} is based on the observation that for small $t$, $f_{\pm}^t=\mathrm{id}+t f_{\pm}$ are diffeomorphisms and $f_+^t(\Omega)=f_-^t(\Omega)$ because $f_+|_{\partial\Omega}=f_-|_{\partial\Omega}$. Hence, by the change-of-variables formula,
\[\int_{\Omega}\det(I+t f_+'(x))\,dx=\mathrm{Vol}(f_+^t(\Omega))=\mathrm{Vol}(f_-^t(\Omega))=\int_{\Omega}\det(I+t f_-'(x))\,dx.\]
Since both sides are polynomials in $t$, comparing the coefficients of $t^n$ gives \eqref{e1}. To make this rigorous, one must handle issues such as why $f_{\pm}^t(\partial\Omega)=\partial f_{\pm}^t(\Omega)$ and why $\partial f_+^t(\Omega)=\partial f_-^t(\Omega)$ implies $f_+^t(\Omega)=f_-^t(\Omega)$; these are addressed by “a careful point set analysis”. The same paragraph notes an advantage: the argument only requires $f_{\pm}$ to be $C^1$.

Another proof approach (Kulpa) defines $I(h^1,\dots,h^n)=\int_{\Omega}\det h'(x)\,dx$ and shows
\[I(f_+^1,f_+^2,\ldots,f_+^n)=I(f_-^1,f_+^2,\ldots,f_+^n)\tag{\ref{e3}}\]
using Fubini’s theorem and coordinate projections $\Pi_i$. Then \eqref{e1} follows by applying \eqref{e3} successively to replace the remaining columns of $\det f_+'(x)$ by those of $\det f_-'(x)$. A further cited route proves \eqref{e1} by using that $L(P)=\det P$ is a null Lagrangian, so the energy $\int_{\Omega}L(w'(x),w(x),x)\,dx$ “depends only on boundary conditions”.

Krylov's proof of the main theorem is based on the observation that for small $t$, $f_{\pm}^t=\mathrm{id}+t f_{\pm}$ are diffeomorphisms and $f_+^t(\Omega)=f_-^t(\Omega)$ because $f_+|_{\partial\Omega}=f_-|_{\partial\Omega}$. Hence, by the change-of-variables formula,
\[\int_{\Omega}\det(I+t f_+'(x))\,dx=\mathrm{Vol}(f_+^t(\Omega))=\mathrm{Vol}(f_-^t(\Omega))=\int_{\Omega}\det(I+t f_-'(x))\,dx.\]
Since both sides are polynomials in $t$, comparing the coefficients of $t^n$ gives
\begin{equation}
    \int_{\Omega} \det f_+' (x) \mathd x = \int_{\Omega} \det f_-' (x) \mathd
    x \text{.} \label{e1}
  \end{equation}
To make this rigorous, one must handle issues such as why $f_{\pm}^t(\partial\Omega)=\partial f_{\pm}^t(\Omega)$ and why $\partial f_+^t(\Omega)=\partial f_-^t(\Omega)$ implies $f_+^t(\Omega)=f_-^t(\Omega)$; these are addressed by “a careful point set analysis”. The same paragraph notes an advantage: the argument only requires $f_{\pm}$ to be $C^1$.

Another proof approach (Kulpa) defines $I(h^1,\dots,h^n)=\int_{\Omega}\det h'(x)\,dx$ and shows
\begin{equation}
  I (f^1_+, f^2_+, \ldots, f_+^n) = I (f^1_-, f^2_+, \ldots, f_+^n) \label{e3}
\end{equation}
using Fubini’s theorem and coordinate projections $\Pi_i$. Then the equation above gives \eqref{e1} by applying the preceding identity successively to replace the remaining columns of $\det f_+'(x)$ by those of $\det f_-'(x)$. A further cited route proves the equation above by using that $L(P)=\det P$ is a null Lagrangian, so the energy $\int_{\Omega}L(w'(x),w(x),x)\,dx$ “depends only on boundary conditions”.

Question: Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\). Let \(f_{\pm}:\overline{\Omega}\to\mathbb{R}^n\) be \(C^2\) maps such that their boundary values agree, i.e. \(f_+|_{\partial\Omega}=f_-|_{\partial\Omega}\). Here \(f_\pm'(x)\) denotes the Jacobian matrix of \(f_\pm\) at \(x\), and \(\det f_\pm'(x)\) its Jacobian determinant. Which statement holds for every such pair of maps?

\[\int_{\Omega} \det f_+'(x)\,dx = \int_{\Omega} \det f_-'(x)\,dx.\]

Choices:
B. \[\det f_+'(x)=\det f_-'(x)\quad\text{for almost every }x\in\Omega.\]
C. \[\text{If }f_-\text{ is constant on }\overline\Omega,\text{ then }\int_{\Omega} \det f_+'(x)\,dx=0.\]
D. \[\int_{\Omega} \bigl|\det f_+'(x)\bigr|\,dx = \int_{\Omega} \bigl|\det f_-'(x)\bigr|\,dx.\]
E. \[\int_{\Omega} \det\!igl(I+f_+'(x)\bigr)\,dx = \int_{\Omega} \det\!igl(I+f_-'(x)\bigr)\,dx.\]

The stem does not state or strongly hint at the specific conclusion; it only gives the boundary-agreement hypothesis and asks which identity follows. The correct answer is not leaked without knowing the underlying theorem.

This is essentially a direct restatement/application of the null-Lagrangian theorem for the Jacobian determinant: same boundary trace implies equality of the integrals of det Df. The MCQ mostly asks the student to recall that exact conclusion.

Some reasoning is needed to distinguish the exact theorem from nearby variants (stronger, weaker, or false modifications), but for a student who recognizes the theorem, the correct choice is fairly immediate.

The distractors are strong and mathematically meaningful: one is weaker-but-true, one is an overgeneralized small-t variant, one confuses signed and absolute determinant, and one is a plausible but false determinant modification. They target realistic failure modes.

A mathematically well-constructed MCQ with strong distractors and no answer leakage, but it is close to a direct theorem restatement and therefore only moderately tests generative reasoning.


Harmonic functions have long been a central object of study in geometric analysis. Among the most celebrated and foundational results in the field is Yau’s Liouville theorem \cite{Yau75}, which asserts that on a complete Riemannian manifold with nonnegative Ricci curvature, any bounded harmonic function must be constant. This theorem not only captures a fundamental rigidity phenomenon, but also inaugurated a far-reaching program aimed at understanding how analytic growth conditions reflect the large-scale geometry of the underlying manifold.

\medskip
Since Yau’s work, the study of harmonic functions and spaces of harmonic functions has developed extensively. Notable contributions include the work of Li and Tam \cite{LiTam1989}, as well as Colding and Minicozzi \cite{ColdingMinicozzi97, ColdingMinicozziAnnals}, who investigated the structure and finite dimensional properties of spaces of harmonic functions of polynomial growth. Further important developments include results of Ni and Tam \cite{NiTam2003} on pluriharmonic functions, and more recently Liu’s work on holomorphic functions via three--circle theorems \cite{LiuThreeCircle}. Collectively, these results demonstrate that under nonnegative Ricci curvature, harmonic functions exhibit rigidity properties closely resembling those of the Euclidean setting.

\medskip
By contrast, Liouville-type phenomena for higher-order elliptic equations have received comparatively little attention. This gap is striking given the intrinsic analytic and geometric relevance of higher-order operators, such as the Paneitz operator in conformal geometry or the operators arising in the study of polyharmonic flows for curves. Aside from a few notable exceptions, there are very few results addressing Liouville-type properties in this broader context. Among these, we mention the works of Branding \cite{Branding18, Branding2021}, where Liouville-type theorems are obtained for biharmonic \emph{maps} under smallness assumptions on certain Sobolev norms.

\medskip
Recently, the authors proved the following result (see \cite{BravoCortissoz2025}, as well as the related work \cite{WangZhu2025}).

\medskip
Recently, the authors proved the following result (see \cite{BravoCortissoz2025}, as well as the related work \cite{WangZhu2025}).

The purpose of the present paper is to extend Theorem~\ref{thm:biharmonic} to the general polyharmonic setting. Recall that a $k$-polyharmonic function on a Riemannian manifold $(M,g)$ is a function $u\in C^{2k}(M)$ satisfying
\[
\Delta^k u = 0,
\]
where $\Delta$ denotes the Laplacian.

\begin{theorem}
\label{thm:main}
Let $(M,g)$ be a complete Riemannian manifold with
$\mbox{Ric}_g\geq 0$.
Then any $k$-polyharmonic function $u\in C^{2k}(M)$ satisfying
\[
u=o(r^{2(k-1)}) \quad \mbox{as}\quad r\rightarrow \infty,
\]
where $r=d(p,x)$, is $(k-1)$-polyharmonic.
In particular, any polyharmonic function of sublinear growth must be constant.
\end{theorem}

\begin{lemma}[Cutoff with explicit bounds]\label{lem:cutoff}
Let $(M^n,g)$ be a complete Riemannian manifold
with $\mbox{Ric}_g\geq 0$
Set $B_\rho:=B_\rho(p)$. Let $r$ be such that and for $0<\frac{1}{8}R\leq r\leq R$.
Then there exists $\chi\in C_c^\infty(B_R)$ such that
\[
0\le \chi\le 1,\qquad
\chi\equiv 1\ \text{on } B_r,\qquad
\mbox{supp}\left(\chi\right)\subset B_R,
\]
and
\[
\left|\nabla \chi\right|\le \frac{C(n)}{R-r},
\quad
|\Delta\chi|\le \frac{C(n)}{(R-r)^{2}}
\quad \text{on}\quad B_R.
\]
\end{lemma}
\begin{proof}
 Take $\chi=\phi$, where $\phi$ is as in Corollary 2.3 in
 \cite{BianchiSetti2018}. Notice that given $r$, we can write $R=\gamma r$ with $\gamma>1$, and in this case,
 we obtain the following estimates for the gradient and Laplacian of the cutoff function.
 \[
 \left|\nabla \chi\right|\leq \frac{C_1}{R}=\frac{\left(\gamma-1\right)C_1}{\left(\gamma-1\right) R}\leq
 \frac{7C_1}{R-r},
 \]
 where we used that $r\geq \frac{1}{8}R$.

\begin{lemma}\label{cor:L2-Estimate Laplacian Biharmonic}
Let $(M,g)$ be an $n$–dimensional complete Riemannian manifold with $\mbox{Ric}_g\ge 0$.
Let $a>0$ and let $u\in C^\infty(B_{a})$ be Biharmonic, i.e.\ $\Delta^2 u=0$ on $B_{a}$.
Then, there is a constant $C>0$ such that for every $s,r\in(0,a)$, with $0<\frac{1}{8}r<s < r<a$
\begin{equation}\label{eq:L2-HF-polh}
\int_{B_s}|\Delta u|^2 dV \leq \frac{C}{(r-s)^4}\int_{B_{r}} u^2 dV.
\end{equation}

\subsection{Proof of Main Theorem \ref{thm:main}}
\label{sect:proof_main_thm}
We are ready to prove our main result. Let $u$ be a $k$-polyharmonic function.
By Corollary \ref{cor:L2-Estimate Laplacian k-1} we can estimate
\begin{equation*}
\label{ineq:Cacciopoli_laplacian}
\int_{B_{r}}|\Delta^{k-1} u|^2\,dV
\leq \frac{C'}{r^{4(k-1)}}\int_{B_{2^{k-1}r}}u^2\,dV.
\end{equation*}
By the volume doubling property (which holds if $\mbox{Ric}\geq 0$), we have
\begin{eqnarray*}
\frac{1}{\mbox{Vol}\left(B_{r}\right)}\int_{B_{r}}|\Delta^{k-1} u|^2\,dV
&\leq& \frac{C'}{r^{4(k-1)}}\frac{1}{\mbox{Vol}\left(B_{2^{k-1}r}\right)}\int_{B_{2^{k-1}r}}u^2\,dV,
\end{eqnarray*}
and hence, since $\Delta^{k-1} u$ is harmonic, then $|\Delta^{k-1} u|$ is subharmonic, by the mean value inequality (Lemma \ref{lemma:mean_value_ineq}),
\begin{eqnarray*}
\sup_{x\in B_{r/2}}\left|\Delta^{k-1} u\right|^2 &\leq&
\frac{C''}{r^{4(k-1)}}\frac{1}{\mbox{Vol}\left(B_{2^{k-1}r}\right)}\int_{B_{2^{k-1}r}}u^2\,dV\\
&\leq& \frac{C''}{r^{4(k-1)}}\sup_{x\in B_r\left(p\right)}\left|u\left(x\right)\right|^2
=\frac{1}{r^{4(k-1)}}o\left(r^{4(k-1)}\right)
\end{eqnarray*}
and hence if $u=o(r^{2(k-1)})$, letting $r\rightarrow \infty$, it
follows that
\[
\Delta^{k-1} u \equiv 0,
\]
i.e., $u$ is $(k-1)$-polyharmonic. For the last statement, let $u$ be a polyharmonic function of
sublinear growth, i.e.
\[
\sup_{B_r}|u| = o(r) \quad \text{as } r\to\infty.
\]
For every $k\ge2$ we have
\[
\frac{\sup_{B_r}|u|}{r^{2(k-1)}}
= o\big(r^{1-2(k-1)}\big) \;\longrightarrow\;0.
\]
In particular $u = o\big(r^{2(k-1)}\big)$ for each $k\ge2$.
Assume that $u$ is $k$–polyharmonic for some $k\ge2$. By the first
part of the theorem, $u$ is then $(k-1)$–polyharmonic. Iterating
this argument for $k,k-1,\dots,2$, we conclude that $u$ is harmonic
and it is sublinear growth.

As stated in the introduction, inequality \eqref{eq:Cacciopoli_Laplacian}
does not appear in \cite{BravoCortissoz2025}. However,
the estimate can be inferred from the proofs of Lemmas 5 and 6
in \cite{BravoCortissoz2025} as follows. From the proof of the local
energy estimate for the Laplacian, Lemma 5, for a cutoff function
$\varphi\in C_c^\infty(B_{2r})$ satisfying for $0<\frac{1}{8}r<s<r$
\[
\varphi\equiv1 \text{ on }B_s,\quad 0\le\varphi\le1,\quad
|\nabla\varphi|\le \frac{C_0}{r-s}\varphi,\quad
|\Delta\varphi|\le \frac{C_0}{(r-s)^2},
\]
we have
\begin{lemma}
\label{lem:laplacian_estimate}
Let $(M,g)$ be an $n$–dimensional complete Riemannian manifold with $\mathrm{Ric}_g\ge 0$. Let $B_{\rho}\colon = B_{\rho}\left(p\right)$.
Fix $a>0$ and let $u\in C^\infty(B_{a})$ be biharmonic, i.e.\ $\Delta^2 u=0$ on $B_{2a}$. Then there exists a constant $C=C(n)>0$ such that, for every pair of radii $0<\dfrac{1}{8}r<s<r\le a$,
\[
\int_{B_s}|\Delta u|^2\,dV
\leq \frac{C}{(r-s)^2}\int_{B_r}|\nabla u|^2\varphi^2\,dV
\;+\; \frac12\int_{B_r} |\Delta u|^2\,dV.
\]
\end{lemma}

From the proof of Lemma 6, the Cacciopoli inequality
for a biharmonic function, for a cutoff
$\varphi\in C_c^\infty(B_{2r})$ such that $0\leq \varphi\leq 1$,
$\varphi\equiv 1$ on $B_s$, and for $\dfrac{1}{8}r<s<r$, it also
satisfies
\[
\left|\nabla \varphi\right|^2\leq \frac{C}{\left(r-s\right)^2}\varphi\quad
 \text{ and } \quad
 \frac{\left|\Delta\left(\varphi^4\right)\right|^2}{\varphi^4}\leq \frac{C}{\left(r-s\right)^4},
\]
we have
\begin{lemma}
\label{lem:Caccio-1-biharmonic-Br}
Let $(M,g)$ be a smooth Riemannian manifold
such that $\mbox{Ric}\geq 0$.
Let $0<\dfrac{1}{8}r<s<r<\infty$
and let $B_\rho:=B_\rho(p)$. If $u\in C^\infty(B_{2r})$ is \emph{biharmonic}, i.e.\ $\Delta^2 u=0$ in $B_{2r}$, then there exists a constant $C=C(n)>0$ such that
\begin{equation*}\label{eq:Caccio-int-Br}
\int_{B_r} |\nabla u|^2\varphi^2\,dV \;\le\; \frac{C}{(r-s)^2}\int_{B_r} u^2\,dV.
\end{equation*}
\end{lemma}

\begin{equation}
\label{eq:Cacciopoli_Laplacian}
\int_{B_s}|\Delta u|^2\,dV \leq \frac{C}{(r-s)^4}\int_{B_r}u^2\,dV,
\end{equation}

\begin{theorem}
\label{thm:biharmonic}
Let $(M,g)$ be a complete Riemannian manifold with a pole and such that
$\mbox{Ric}_g\geq 0$.
Then any biharmonic function $u\in C^{4}(M)$ of subquadratic growth is harmonic.
In particular, any biharmonic function of sublinear growth must be constant.
\end{theorem}

The purpose of the present paper is to extend Theorem~\ref{thm:biharmonic} to the general polyharmonic setting. Recall that a $k$-polyharmonic function on a Riemannian manifold $(M,g)$ is a function $u\in C^{2k}(M)$ satisfying
\[
\Delta^k u = 0,
\]
where $\Delta$ denotes the Laplacian.

\medskip
Our main result is the following.

\begin{theorem}
\label{thm:main}
Let $(M,g)$ be a complete Riemannian manifold with
$\mbox{Ric}_g\geq 0$.
Then any $k$-polyharmonic function $u\in C^{2k}(M)$ satisfying
\[
u=o(r^{2(k-1)}) \quad \mbox{as}\quad r\rightarrow \infty,
\]
where $r=d(p,x)$, is $(k-1)$-polyharmonic.
In particular, any polyharmonic function of sublinear growth must be constant.
\end{theorem}

\medskip
As is well known, Theorem~\ref{thm:main} has long been established in the Euclidean setting. In $\mathbb{R}^n$, the critical growth rate that separates genuinely $k$-polyharmonic behavior from lower-order behavior is precisely $r^{2(k-1)}$, as follows from Almansi’s decomposition. This shows that Theorem~\ref{thm:main} is sharp and constitutes a natural geometric extension of the classical Euclidean theory.

\medskip
We now briefly describe the main analytic difficulty underlying the proof of Theorem~\ref{thm:main}. We begin by recalling the strategy used in the proof of Theorem~\ref{thm:biharmonic}. In \cite{BravoCortissoz2025}, the argument relied on an energy estimate for the $L^2$-norm of $\Delta u$ together with a Caccioppoli-type inequality,
\begin{equation}
\label{eq:Cacciopoli}
\int_{B_s}|\nabla u|^2\,dV \leq \frac{C}{(r-s)^2}\int_{B_r}u^2\,dV,
\end{equation}
valid for both harmonic and biharmonic functions. This estimate was then used to derive
\begin{equation}
\label{eq:Cacciopoli_Laplacian}
\int_{B_s}|\Delta u|^2\,dV \leq \frac{C}{(r-s)^4}\int_{B_r}u^2\,dV,
\end{equation}
which, combined with a mean value inequality, yields the desired rigidity.

\medskip
At first glance, these arguments suggest the possibility of an inductive approach to Theorem~\ref{thm:main}. However, the proof of \eqref{eq:Cacciopoli} is fundamentally based on the Bochner formula, and this approach does not extend in any straightforward way to polyharmonic functions beyond the biharmonic case. To overcome this obstacle, we establish a direct analogue of \eqref{eq:Cacciopoli_Laplacian} for \emph{arbitrary polyharmonic functions}. This new estimate is the most delicate part of the argument and forms the analytic backbone of the inductive procedure. Although the biharmonic case serves as the base step, the proof requires a genuinely new estimate that may be of independent interest.

\medskip
Taken together with Yau’s theorem and our previous work on biharmonic functions, the results of this paper complete, on manifolds with nonnegative Ricci curvature, a natural hierarchy of Yau--Liouville type theorems for the Laplacian and its iterates. They also raise the question of whether similar rigidity phenomena persist for more general higher-order elliptic operators, as they do in the Euclidean setting.

\medskip
Beyond its analytic content, Theorem~\ref{thm:main} may be interpreted as a rigidity principle for complete manifolds with nonnegative Ricci curvature. It shows that, at the level of critical growth, such manifolds are analytically indistinguishable from Euclidean space with respect to iterates of the Laplacian. In particular, the behavior of polyharmonic function in
nonnegative Ricci curvature seems to mirror Almansi’s decomposition in $\mathbb{R}^n$.
\subsection{}

The interplay between the behavior of polyharmonic functions and curvature remains largely unexplored and may reveal further rigidity phenomena. As shown in this paper, when $\mbox{Ric}_g\geq 0$, the qualitative behavior closely parallels that of Euclidean space. In contrast, in negatively curved settings the situation is markedly different. In particular, the authors showed in \cite{BravoCortissoz} that there exist curvature regimes in which every bounded biharmonic function must be harmonic, while nonconstant bounded harmonic functions still exist. This naturally leads to the question of how $k$-polyharmonic functions behave in such settings.

\subsection{}

In a previous version of this manuscript, we imposed one of the following additional assumptions:
\begin{itemize}
\item[(H1)] the geodesic spheres centered at $p$ are mean-convex with respect to the radial vector field, or
\item[(H2)] the square of the distance function from $p$, denoted by $d$, is convex.
\end{itemize}
Either assumption yields a lower bound for the Laplacian of the distance function, which is needed to construct suitable cutoff functions. However, we later observed that the condition $\mbox{Ric}_g\geq 0$, together with the existence of a pole, already implies (H1), rendering it redundant (see \cite{BravoCortissoz2025}). Subsequently, inspired by the recent work \cite{WangZhu2025} and with the generous assistance of Professor Wang, we adopted the cutoff technology developed in \cite{BianchiSetti2018}, which requires only the assumption $\mbox{Ric}_g\geq 0$. We are grateful to Professors Wang and Zhu for their help.

\subsection{Organization of the paper}
The material of this paper is organized as follows. In Section
\ref{section:technical}, we collect some facts about cutoff functions
and the hole-filling technique needed in the proof of the main
$L^2$-estimates needed in the proof of our main result. These
$L^2$ estimates are presented and proved in Section \ref{sect:L2estimates}.
Finally, Theorem \ref{thm:main} is proved in Section \ref{sect:proof_main_thm}.

To prove Theorem~\ref{thm:main}, the introduction proposes an inductive strategy inspired by the biharmonic case (Theorem~\ref{thm:biharmonic}). In \cite{BravoCortissoz2025}, the biharmonic argument uses an energy estimate for the $L^2$-norm of $\Delta u$ together with a Caccioppoli-type inequality
\[
\int_{B_s}|\nabla u|^2\,dV \leq \frac{C}{(r-s)^2}\int_{B_r}u^2\,dV,
\]
which then yields
\[
\int_{B_s}|\Delta u|^2\,dV \leq \frac{C}{(r-s)^4}\int_{B_r}u^2\,dV.
\]
Combined with a mean value inequality, this gives the rigidity conclusion in the biharmonic case.

For Theorem~\ref{thm:main}, the text notes that while this suggests induction, the proof of the basic Caccioppoli inequality is “fundamentally based on the Bochner formula,” and “does not extend in any straightforward way to polyharmonic functions beyond the biharmonic case.” The stated remedy is to “establish a direct analogue of \eqref{eq:Cacciopoli_Laplacian} for \emph{arbitrary polyharmonic functions}.” This new $L^2$ estimate is described as “the most delicate part of the argument” and as “the analytic backbone of the inductive procedure,” with the biharmonic case as the base step but requiring “a genuinely new estimate.”

The introduction also indicates the technical tools used to implement this: cutoff-function technology under $\mathrm{Ric}_g\ge0$ (adopting the approach of \cite{BianchiSetti2018}) and the “hole-filling technique,” and it outlines where the proof components appear: cutoff/hole-filling facts in Section~\ref{section:technical}, the main $L^2$ estimates in Section~\ref{sect:L2estimates}, and the proof of Theorem~\ref{thm:main} in Section~\ref{sect:proof_main_thm}.

To prove Theorem~\ref{thm:main}, the introduction proposes an inductive strategy inspired by the biharmonic case. We first prove the following theorem.
\begin{theorem}
\label{thm:biharmonic}
Let $(M,g)$ be a complete Riemannian manifold with a pole and such that
$\mbox{Ric}_g\geq 0$.
Then any biharmonic function $u\in C^{4}(M)$ of subquadratic growth is harmonic.
In particular, any biharmonic function of sublinear growth must be constant.
\end{theorem}
In Bravo, Cortissoz, \emph{TITLE UNKNOWN} (2025), the biharmonic argument uses an energy estimate for the $L^2$-norm of $\Delta u$ together with a Caccioppoli-type inequality
\[
\int_{B_s}|\nabla u|^2\,dV \leq \frac{C}{(r-s)^2}\int_{B_r}u^2\,dV,
\]
which then yields the estimate
\begin{equation}
\label{eq:Cacciopoli_Laplacian}
\int_{B_s}|\Delta u|^2\,dV \leq \frac{C}{(r-s)^4}\int_{B_r}u^2\,dV,
\end{equation}
Combined with a mean value inequality, this gives the rigidity conclusion in establishing the main theorem.

For Theorem~\ref{thm:main}, the text notes that while this suggests induction, the proof of the basic Caccioppoli inequality is “fundamentally based on the Bochner formula,” and “does not extend in any straightforward way to polyharmonic functions beyond the biharmonic case.” The stated remedy is to “establish a direct analogue of the equation above for \emph{arbitrary polyharmonic functions}.” This new $L^2$ estimate is described as “the most delicate part of the argument” and as “the analytic backbone of the inductive procedure,” with the base step provided by the theorem above but requiring “a genuinely new estimate.”

The introduction also indicates the technical tools used to implement this: cutoff-function technology under $\mathrm{Ric}_g\ge0$ (adopting the approach of \cite{BianchiSetti2018}) and the “hole-filling technique,” and it outlines where the proof components appear: cutoff/hole-filling facts appear later, the main $L^2$ estimates are proved later, and the proof of Theorem~\ref{thm:main} appears later.

Question: Let $(M,g)$ be a complete Riemannian manifold that admits a pole $p$ (that is, the exponential map $\exp_p:T_pM\to M$ is a global diffeomorphism) and satisfies $\mathrm{Ric}_g\ge 0$. Let $r(x)=d(p,x)$. A biharmonic function is a function $u\in C^4(M)$ with $\Delta^2 u=0$. A function has subquadratic growth if $u(x)=o(r(x)^2)$ as $r(x)\to\infty$, and sublinear growth if $u(x)=o(r(x))$ as $r(x)\to\infty$. Which statement holds for every such biharmonic function?

Every biharmonic function $u\in C^4(M)$ with subquadratic growth is harmonic, i.e. $\Delta u=0$ on $M$. In particular, every biharmonic function of sublinear growth is constant.

Choices:
B. Every biharmonic function $u\in C^4(M)$ with subquadratic growth is constant on $M$. In particular, every biharmonic function of sublinear growth is constant.
C. Every biharmonic function $u\in C^4(M)$ with sublinear growth is constant on $M$.
D. Every biharmonic function $u\in C^4(M)$ with quadratic growth, i.e. $u(x)=O(r(x)^2)$ as $r(x)\to\infty$, is harmonic, i.e. $\Delta u=0$ on $M$. In particular, every biharmonic function of sublinear growth is constant.
E. Every biharmonic function $u\in C^4(M)$ with subquadratic growth is harmonic, i.e. there exists a constant $C=C(u)$ such that $\Delta u\equiv C$ on $M$. In particular, every biharmonic function of sublinear growth is constant.

dropped subquadratic_to_harmonic implication and kept only the stated corollary for sublinear growth

little_o threshold replaced by big_O boundary case

exact vanishing of Laplacian weakened to constant Laplacian

The stem gives geometric and growth-condition definitions but does not explicitly state the theorem’s conclusion. There is no direct answer leakage from the wording of the prompt itself.

The correct option is essentially a verbatim theorem statement under the stated hypotheses, including its corollary. This makes the item largely a recall/restatement question rather than a non-tautological inference task.

There is some reasoning pressure because the alternatives differ in subtle but meaningful ways: harmonic vs constant, subquadratic vs sublinear, little-o vs big-O, and zero Laplacian vs constant Laplacian. Still, the item mainly tests recognition of the exact theorem.

The distractors are plausible and mathematically targeted: one is too strong, one is weaker but true, one changes the growth threshold, and one weakens the conclusion incorrectly. These reflect common theorem-misremembering or overgeneralization errors.

A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it is quite tautological because the correct option is essentially the theorem itself.


\begin{thm}\label{ulimitatoCupini}
    Let $f = f(x, \xi)$ be the integrand defined at \eqref{integrand} and assume that $\omega\in L^t(\Omega)$, with $\left(\frac{\overline{p}^*}{p_n} \right)' < t \le \infty$,
where $\bar{p}^*$ is the Sobolev exponent of $\bar{p}$, with $\bar{p}$ defined at \eqref{definizionePi}.
Then, every local minimizer $u$ of \eqref{functional} is locally bounded in $\Omega$.
\end{thm}

\begin{thm}\label{thmBfinito}
Let $\omega \in  W_{\mathrm{loc}}^{1,\frac{\mathbf{p}+2}{\mathbf{p}+1}}(\Omega) $ and let $u \in {W_{\mathrm{loc}}^{1,\mathbf{p}}}(\Omega)$ be a local minimizer of \eqref{functional} under assumptions \eqref{A2}--\eqref{A4}, with exponents $p_i\geq 2, \forall i=1,\dots,n $, such that
\begin{equation}\label{Ipotesip}
p_n <
\begin{cases}
    \min \Big\{ \dfrac{\overline{p}^*}{t'} ,p_1 +2 \Big\} \, &\text{if } \overline{p} < n,  \\
     p_1 +2 \quad &\text{otherwise},
    \end{cases}
\end{equation}
and with a function $g \in {L^{r}_{\mathrm{loc}}(\Omega)}$, where $r$ satisfies the condition
\begin{equation}\label{Ipotesir}
   r> p_n + 2 .
\end{equation}
Then,
\begin{equation*}
{V_{p_i}(u_{x_i}) \in W^{1,2}_{\mathrm{loc}}(\Omega)} \qquad\forall i=1,\dots,n,
\end{equation*}
and the following estimates
\begin{align*}
    &\sum_{i=1}^{n} \left(  \int_{B_{R/4}} |u_{x_i}|^{p_i +2 } dx \right) \notag\\
    &\leq c  \left( 1+\sum_{i=1}^{n} \left(\Vert u_{x_i} \Vert_{L^{p_i}(B_R)} + {\Vert \omega_{x_i} \Vert_{L^\frac{p_i+2}{p_i+1}(B_{R})}} \right) +\Vert \omega\Vert_{L^t(B_R)}+ \Vert g \Vert_{L^r (B_{R})} \right)^\sigma
\end{align*}
and
\begin{align*}
        &\sum_{i=1}^{n}  \left( \int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2}dx \right) \notag\\
         &\leq c  \left( 1+\sum_{i=1}^{n} \left(\Vert u_{x_i} \Vert_{L^{p_i}(B_R)} + {\Vert \omega_{x_i} \Vert_{L^\frac{p_i+2}{p_i+1}(B_{R})}} \right) +\Vert \omega\Vert_{L^t(B_R)}+ \Vert g \Vert_{L^r (B_{R})} \right)^\sigma
\end{align*}
hold for every pair of concentric balls $B_{R/4} \subset B_{R} \Subset \Omega$, where $c = c(n,p_i,\lambda, \Lambda,R, \Vert u \Vert_{\infty})$ and $\sigma= \sigma (n,p_i, r)$ are positive constants.
\end{thm}

\begin{align}\label{Stimacompattauxixjpi+2}
        \sum_{i=1}^{n}   \left( \int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2} dx \right) & \leq c  \left(1+ \sum_{i=1}^{n} \left(\Vert u_{x_i} \Vert_{L^{p_i}(B_R)}+{\Vert \omega_{x_i} \Vert_{L^{\frac{p_i+2}{p_i+1}} (B_R) }} \right) + \Vert g \Vert_{L^r (B_{R})} \right)^\sigma,
\end{align}
which concludes the proof.
\end{proof}
\section{Approximation}\label{Appro}

In order to perform the approximation argument, let us fix a non-negative smooth kernel $\phi \in \mathcal{C}^\infty_0(B_1(0))$ such that $\int_{B_1(0)} \phi =1$ and consider the corresponding family of mollifiers $(\phi_k)_{k >0}$. We set, for a given ball $B_R \Subset \Omega$ ,
$$a_i^k=a_i * \phi_k, \quad \omega^k= \omega * \phi_k,$$
\begin{equation}\label{gk}
g_k(x)=\underset{i}{\max} \{ |D a_i^k (x)|\}
\end{equation}
and, for $x \in B_R$, we define
\begin{equation}\label{fvarepsilon}
f^k (x,\xi)= \sum_{i=1}^{n} a_i^k(x)|\xi|^{p_i} ,
\end{equation}
 for every $k < \text{dist}(B_R,\Omega)$. We shall use the following $$\mathcal{F}^{k}(v,B_R):=\int_{_{B_R} }\left( f^{k}(x, Dv)-\omega(x)v(x) \right) dx$$ and
$$\mathcal{F}^{k}_{\varepsilon}(v,B_R):= \, \int_{B_R} \left( f^k(x,Dv)+ \varepsilon(1+ |Dv|^2)^{\frac{p_n}{2}}- \omega^k(x)v(x) \right) dx.$$
\begin{flushleft}
One can easily check that $f^k(x, \xi)$ satisfies assumptions \eqref{A2}--\eqref{A3} and \eqref{A'4}, with $K = \sup_{B_R} |g_k(x)| $ that is finite since $a^k_i(x) \in W^{1, \infty}(B_R)$ and then $g_k \in L^{\infty}(B_R)$.\\
\end{flushleft}

We are in position to prove our main result.

\begin{proof}[\textit{{Proof of Theorem~\ref{thmBfinito}}}]
We consider the variational problems
\begin{equation}\label{Pfj}
    \inf \left\{  \int_{B_R} \left( f^k(x,Dv)+ \varepsilon(1+ |Dv|^2)^{\frac{p_n}{2}} -\omega^k(x)v(x)\right) dx \ : \ v \in W^{1,p_n}_0(B_R)+u^\eta \right\},
\end{equation}
where $$u^\eta = u * \varsigma_\eta$$ is the mollification of the local minimizer $u$ of \eqref{functional}, for a sequence of mollifier $\varsigma_n$.\\
It is well known that, by the direct methods of the calculus of variations, there exists a unique solution $u^\eta_{k,\varepsilon} \in  W^{1,p_n}_0(B_R)+u^\eta $ of the problem \eqref{Pfj}. Since the integrand $$f^k(x,Dv)+ \varepsilon(1+ |Dv|^2)^{\frac{p_n}{2}}-\omega^k(x)v(x)$$ satisfies the assumptions of  Theorem \ref{thm2Mascolo}, we have that $$V_{p_i}((u^\eta_{k,\varepsilon})_{x_i}) \in W_{\mathrm{loc}}^{1,2}(B_R) \quad \forall i=1,...,n.$$
Hence we are legitimate to use estimate \eqref{uxistima} of Theorem \ref{AppThm}, to obtain that
\begin{align}
   & \sum_{i=1}^{n}  \int_{B_{R/4}}|(u^\eta_{k,\varepsilon})_{x_i}|^{p_i +2 } dx \notag \\
    & \leq c \left(1+ \sum_{i=1}^{n} \left(\int_{B_R}\, \lvert(u^\eta_{k,\varepsilon})_{x_i} \rvert^{p_i} \ dx+  \Vert \omega^k_{x_i} \Vert_{L^\frac{p_i+2}{p_i+1}(B_{R})} \right)  + \Vert g_k \Vert_{L^r (B_{R})} \right)^\sigma \notag \\
     &  \leq c(\lambda) \left(1+ \sum_{i=1}^{n} \left( \int_{B_R} a_i^k(x)\, \lvert (u^\eta_{k,\varepsilon})_{x_i}\rvert^{p_i} \ dx +  \Vert \omega^k_{x_i} \Vert_{L^\frac{p_i+2}{p_i+1}(B_{R})} \right) + \Vert g_k \Vert_{L^r (B_{R})} \right)^\sigma, \label{der}
\end{align}

\begin{lem}\label{LemmaTroisi}
    Let $E \subset \mathbb{R}^n$ be a bounded open set and consider $u \in W^{1, \mathbf{p}}(E)$, with $  p_i \geq 1$, for all $i=1, ..., n$.\\
    If $\overline{p} \le n$, then there exists a positive constant $\gamma_1$ depending on $n, p_i$ and, only in the case $\overline{p}=n$, also on $\overline{p}^*$ and $E$, such that
    \begin{align*}
        \lvert \lvert u \rvert \rvert_{L^{\overline{p}^*}(E)} \leq \gamma_1  \left[ \sum_{i=1}^{n} \lvert \lvert u_{x_i} \rvert \rvert_{L^{p_i}(E)} + \lvert \lvert u \rvert \rvert_{L^1(E)} \right].
    \end{align*}
      If $\overline{p} > n$,  there exists a positive constant $\gamma_2$ depending only on $ n, p_i$ and $E$, such that
    \begin{align*}
        \lvert \lvert u \rvert \rvert_{L^{\infty}(E)} \leq \gamma_2  \left[ \sum_{i=1}^{n} \lvert \lvert u_{x_i} \rvert \rvert_{L^{p_i}(E)} + \lvert \lvert u \rvert \rvert_{L^1(E)} \right].
    \end{align*}
\end{lem}

\section{A regularized problem}\label{PreREG}
In this section we prove a regularity result for more regular problems with respect to \eqref{functional} that will be needed in the approximation procedure. More precisely, for $\varepsilon \geq 0$ we consider
\begin{equation}\label{Eqconepsilon}
 f_\varepsilon(x,u, \xi)= \sum_{i=1}^{n} \, \dfrac{b_i(x)}{p_i} \lvert \xi_i \rvert^{p_i} + \varepsilon(1+|\xi|^\frac{p_n}{2})^2  -\overline{\omega}(x)u
\end{equation}
with
$b_i(x):\Omega \to [0,+\infty), \, i=1, \dots, n$, non-negative, Lipschitz continuous coefficients,
i.e., the following conditions hold
\begin{equation}\label{atilde}
L_i= \sup_{x,y \in \Omega, x \neq y} \frac{|b_i(x)-b_i(y)|}{|x-y|}< \infty , \quad i=1,..,n.
\end{equation}
Setting $\tilde{f}(x, \xi)= \sum_{i=1}^{n} \, b_i(x) \lvert \xi_i \rvert^{p_i},$ one can easily check that
\begin{equation}\label{A'4}
    |D_\xi  \tilde{f}(x,\xi)-D_\xi \tilde{f}(y, \xi)| \leq K |x-y|\, \, \sum_{i=1}^{n} |\xi_i|^{p_i-1} \tag{A3'}
\end{equation}
\noindent for a.e.\ $x,y \in \Omega $ and every $\xi \in \mathbb{R}^{ n}$, where $K=K(L_i, p_i)$
.\\
We recall a Lipschitz regularity Theorem for the minimizers of the functional
\begin{equation}\label{functional2}
    \mathcal{F}_\varepsilon (x, Dv) =\int_\Omega f_\varepsilon(x,v, Dv) \, dx,
\end{equation}
that can be deduced by  \cite[Theorem $8.9$]{giusti} in the case $p=p_n$, where $f_\varepsilon$ was defined at \eqref{Eqconepsilon}.
\begin{thm}\label{thm2Mascolo}
    Let $\tilde{f} $ satisfy \eqref{A2}, \eqref{A3} and \eqref{A'4} and $\overline{\omega} \in L^\infty(\Omega)$.
    Let $u_\varepsilon \in W^{1,p_n}_{\mathrm{loc}}(\Omega)$ be a local minimizer of \eqref{functional2}.
Then, $u_\varepsilon \in W^{1,\infty}_{\mathrm{loc}}(\Omega)$
\end{thm}

We shall use the higher differentiability of the minimizers $u_\varepsilon$ of $\mathcal{F}_\varepsilon$ that, as far  we know, is not available in literature and that is contained in the following
{
\begin{lem}\label{LemmaInduzione}
  Let $\tilde{f}$ satisfy \eqref{A2}, \eqref{A3} and \eqref{A'4} for exponents $ p_i \geq 2,  \forall i=1,\dots,n$, and assume {$\overline{\omega} \in L^\infty(\Omega) \cap W^{1,\frac{\mathbf{p}+2}{\mathbf{p}+1}}_{\mathrm{loc}}(\Omega)$. }
    Let  $u_\varepsilon \in W^{1,p_n}_{\mathrm{loc}}(\Omega) $ be a local minimizer of the functional \eqref{functional2}.
    Then,
    we have  $${V_{p_i}((u_\varepsilon)_{x_i})} \in W^{1,2}_{\mathrm{loc}}(\Omega), \quad \forall i =1,\dots, n.   $$

\end{lem}

\begin{lem}\label{D1}
Let $1<p<+\infty$. There exist positive constants $c_1$ and $c_2$, depending only on $n$ and $p$, such that
\begin{center}
{$c_1(|\xi|^{2}+|\eta|^{2})^{\frac{p-2}{2}} \leq \dfrac{|V_{p}(\xi)-V_{p}(\eta)|^{2}}{|\xi-\eta|^{2}} \leq c_2(|\xi|^{2}+|\eta|^{2})^{\frac{p-2}{2}} $},
\end{center}
for any $\xi, \eta \in \mathbb{R}^{n}$, with $\xi \neq \eta$.
\end{lem}

The aim of this paper is to investigate the differentiability properties of locally bounded minimizers of convex functionals, exhibiting an orthotropic structure, depending also on the $x$-variable and on the minimizer itself. More precisely, we consider
\begin{equation}\label{functional}
 \mathcal{F}(u,\Omega):= \, \int_\Omega \left[f(x, Du)-\omega(x)u(x) \right] dx,
\end{equation}
\noindent where $\Omega \subset \mathbb{R}^n$ is an open subset, $n\geq 2$, {$u : \Omega \to \mathbb{R}$}, and $f:\Omega\times\mathbb{R}^{ n}\to [0,+\infty)$ has the following form
\begin{equation}
    f= f(x, \xi)= \sum_{i=1}^{n} \,     \dfrac{a_i(x)}{p_i} \lvert \xi_i \rvert^{p_i}, \label{integrand}
\end{equation}
with $\xi=(\xi_1,\xi_2,...,\xi_n) \in \mathbb{R}^n$.
Here, we assume that $2\leq p_1  \leq \cdots \leq p_n$  and $a_i(x):\Omega \to \mathbb{R}$ are bounded, measurable coefficients such that, for every $i = 1,...,n$,   $$0< \lambda \leq a_i (x) \leq \Lambda  ,$$ for some positive constants $\lambda$, $ \Lambda$ and for a.e.\ $ x \in \Omega$.

One can easily check  that there exist positive constants $ L, l$ depending on $\lambda, \Lambda, p_i$ such that
\begin{equation}\label{A2}
    \langle D_\xi f(x, \xi)-D_\xi f(x, \eta),\xi-\eta\rangle  \geq  l \sum_{i=1}^{n}(|\xi_i|^2+|\eta_i |^2)^\frac{p_i -2}{2}|\xi_i - \eta_i |^2 \tag{A1}
\end{equation}
\begin{equation}\label{A3}
     |D_\xi f(x, \xi)-D_\xi f(x, \eta)| \leq  L \sum_{i=1}^{n}(|\xi_i |^2+|\eta_i |^2)^\frac{p_i -2}{2}|\xi_i - \eta_i | ,\tag{A2}
\end{equation}

\noindent for a.e.\ $x \in \Omega$ and all $\xi, \eta \in \mathbb{R}^{ n}$.
Concerning the dependence on the $x$-variable, we assume that $$a_i(x) \in W^{1,r}_{\mathrm{loc}}(\Omega) \qquad \forall i=1, \dots,n ,$$ so that,
denoting by $g(x)=  \underset{i}{\max} \{|Da_i(x)| \}$, we have that $g(x) \in L_{\mathrm{loc}}^r (\Omega)$ and the following inequality
\begin{equation}\label{A4}
    |D_\xi f(x,\xi)-D_\xi f(y, \xi)| \leq |x-y|\,(g(x)+g(y)) \, \sum_{i=1}^{n} |\xi_i|^{p_i-1} \tag{A3}
\end{equation}
\noindent holds for a.e.\ $x,y \in \Omega$ and every $\xi \in \mathbb{R}^{n}$.

For the exponents $p_i$, we shall denote by $\mathbf{p} = (p_1, p_2, \ldots, p_n)$ and by
\begin{equation*}
    W^{1,\mathbf{p}}_{\mathrm{loc}}(\Omega) = \left\{ u \in W^{1,1}_{\mathrm{loc}}(\Omega) : u_{x_i} \in L^{p_i}_{\mathrm{loc}}(\Omega), \, i = 1, \ldots, n \right\},
    \end{equation*}
the  corresponding anisotropic Sobolev space.\\
We denote by $\overline{p}$ the harmonic average of $p_i$ and by $\overline{p}^*$ the Sobolev conjugate exponent of $\overline{p}$, i.e.
 \begin{align}\label{definizionePi}
        \frac{1}{\overline{p}} = \frac{1}{n} \sum_{i=1}^{n} \frac{1}{p_i}, \qquad \overline{p}^* = \begin{cases}\frac{n \overline{p}}{n- \overline{p}} & \quad \text{if} \ \overline{p} <n,\\
        \text{any exponent in } [1,\infty) & \quad \text{if} \  \overline{p} \ge n.
        \end{cases}
    \end{align}
   For an exponent $1 \le s \le \infty$, we will denote by $s'$ the H\"older conjugate of $s$.\\
In the following, we will assume that the function $\omega(x) $ belongs to the Sobolev space  $W^{1,{\frac{\mathbf{p}+2}{\mathbf{p}+1}}}_{\mathrm{loc}}(\Omega)$, with ${\frac{\mathbf{p}+2}{\mathbf{p}+1}} = (\frac{p_1+2}{p_1+1}, \frac{p_2+2}{p_2+1}, \ldots, \frac{p_n+2}{p_n+1})$.\\

\noindent Let us recall the following definition of local minimizer.
\begin{dfn}
{ \em A function {$u\in W_{\rm loc}^{1,\mathbf{p}}(\Omega)$} is a  local minimizer of
\eqref{functional} if, for every open subset $\tilde{\Omega} \Subset \Omega$, we have
$\mathcal{F}(u;\tilde{\Omega})\le  \mathcal{F}(\varphi;\tilde{\Omega})$
for all $\varphi\in \mathcal{C}_0^{\infty}(\tilde{\Omega})$.}
\end{dfn}
The energy densities $f$ that satisfy anisotropic growth condition fit into the wider context of those with $(p,q)$-growth condition as follows
$$|\xi|^p \leq f(x,\xi) \leq (1 + |\xi|^2)^{\frac{q}{2}}, \qquad1<p<q, $$
where $p= c(p_i)$ and $q= \max \{p_i : i=1, \dots, n \}.$\\
The study of regularity properties of local minimizers to functionals with non-standard growth started with the pioneering papers by Marcellini \cite{MarcelliniEs1, MarcelliniEs2}.
 It is now well understood that, in order to obtain some regularity for the local minimizers of functionals with $(p,q)$-growth, $p<q$,  a restriction between the exponents $p$ and $q$ must be imposed (see the counterexamples in \cite{Giaquinta,MarcelliniEs3}).  Indeed, the gap $\frac{q}{p}$ cannot differ to much from $1$ and sufficient conditions to the regularity can be expressed as
$$\frac{q}{p}\le c(n)\displaystyle \to 1\,\,\textnormal{as}\,\,\,\,n\to \infty,$$
according to \cite{MarcelliniEs1,MarcelliniEs2}.

In recent years there has been a considerable of interest in variational problems with $(p,q)$-growth conditions (see, for example, \cite{Adi,Barone,6,12,SuggeritoRefery,Ming1,Ming2,Ming3,Ming4,Ele1,Ele2,Ele3,25,Hasto,Koch,16} and references therein). It is impossible to give a complete overview on the subject, however we refer the interested reader to the recent survey \cite{surveyMar1,surveyMing}.\\
Compared to the above mentioned papers, the difference with our work lies in the fact that the behavior of our functional at \eqref{functional} depends on the components of the gradient and, although they exhibit $(p,q)$-growth, they need to be treated with appropriate arguments.\\
Actually, the regularity of local minimizers of functionals with orthotropic structure as in \eqref{functional} is a well-studied problem: some results can be found in \cite{6,5,LemmaBrasco,Carozza,17,53,russo}, for the homogeneous case \( \omega = 0 \), and for the non-homogeneous one, we refer to  \cite{BraTau,CMM}.
\\
In the autonomous case, i.e.\ $a_i(x) \equiv 1$,  for all $i=1,...,n$, it has been proved in \cite{Brascop12}, for the sub-quadratic case  with $\omega \in L^{1+ \frac{2}{p_1}}_{\mathrm{loc}}(\Omega)$,
that any local minimizer \( u \in W^{1,\mathbf{p}}_{\text{loc}}(\Omega) \cap L^\infty_{\text{loc}}(\Omega) \)  satisfies
\begin{equation}
    |u_{x_i}|^{\frac{p_i - 2}{2}}  \, u_{x_i} \in W^{1,2}_{\text{loc}}(\Omega), \quad i = 1, \dots,n. \label{diffproperties}
\end{equation}
For the super-quadratic case, Brasco et al.\ \cite{BraTau} consider the following widely degenerate orthotropic functional
$$F(u; \Omega) = \sum_{i=1}^{n} \int_{\Omega} \frac{1}{p_i} \left( |u_{x_i}| - \delta_i \right)^{p_i} \, dx - \int_{\Omega} \omega \,u \, dx, $$
with $\delta_i \ge 0$, $p_i \ge 2$, $ u \in W^{1,\mathbf{p}}_{\text{loc}}(\Omega)\cap L^\infty_{\text{loc}}(\Omega)$ and
$ \omega \in W^{1,\mathbf{p'}}_{\text{loc}}(\Omega), $ $\mathbf{p'}=(p_1', \cdots, p_n')$, where $p_i'$ is the H\"older conjugate of $p_i$.
The authors proved the higher differentiability of the local minimizers, that is
\[
\left( |u_{x_i}| - \delta_i \right)^\frac{p_i}{2}_+ \frac{u_{x_i}}{|u_{x_i}|}  \in W^{1,2}_{\text{loc}}(\Omega), \quad i = 1, \dots, n.
\]
In particular, for local weak solutions of the anisotropic orthotropic $p$-Laplace equation (i.e. the case $\delta_i = 0$), they get the regularity at \eqref{diffproperties}.

 In the paper \cite{16} by Marcellini, it is clearly stated that currently there are few known regularity results for problems of this type, in relation to the properties of the coefficients $a_i(x)$. Indeed, as far as we know, few regularity results are available for the local minimizers of functionals with an orthotropic structure in the so-called non-autonomous case (see \cite{feopass,russo}).
More precisely, in the super-quadratic case it has been recently proved in \cite{russo} that, assuming the condition $p_n < p_1+2$ and Sobolev regularity on the coefficients $a_i(x)$ of the type $W^{1,r}$, with $r > p_n+2$, locally bounded minimizers $u$ of the functional \eqref{functional}, with $\omega  \equiv 0$, satisfy the higher differentiability properties at \eqref{diffproperties}.\\
\noindent Here, the main novelty with respect to the paper \cite{russo} is that we allow the presence of a forcing term $\omega$. We are interested in the conditions that must be imposed on the function $\omega$ in order to achieve analogous higher differentiability results for the solutions. \\
 Actually, our main result consists in proving that a suitable Sobolev regularity of the coefficients $a_i(x)$ and of the forcing term $\omega$ transfers into a higher differentiability of integer order for the local minimizers of \eqref{functional}. Moreover, we will require a weaker differentiability assumption on $\omega$ with respect to the one made in \cite{BraTau}. Indeed,  since $\frac{p_i+2}{p_i+1} < p_i'$, we have the embedding $W^{1, \mathbf{p'}}_{\mathrm{loc}}(\Omega) \hookrightarrow W^{1,{\frac{\mathbf{p}+2} {\mathbf{p}+1}}}_{\mathrm{loc}}(\Omega)$.

We recall that recent studies have shown that the weak differentiability of the map \( D_\xi f(x,\xi) \), whether of integer or fractional order with respect to the \( x \)-variable, is a sufficient condition to achieve higher differentiability (see \cite{cup, Torricelli, 25,40} for the case of Sobolev spaces with integer order and \cite{ 2,15, Grimaldi1, Ipocoana1, Russo} for the fractional one) both in case of standard and non standard growth. \\
In particular, the general requirement for obtaining higher differentiability for local minimizers is a Sobolev-type regularity for the coefficients with an exponent
$r$ that is greater than or equal to the dimension
$n$. When dealing with bounded minimizers, the situation changes, and sufficient assumptions on the Sobolev regularity of the coefficients can be made independently of the dimension (see \cite{CKP, Colombo, cup,25}). Our result follows this approach. Even though the order of summability of the coefficients depends on the dimension $n$, it does so only through the bound that allows us to handle locally bounded minimizers.

{In the statement our main result, we shall use the  auxiliary function $V_{p}(\xi)$  defined by
\begin{equation}\label{DefVp}
    V_{p}(\xi):= |\xi|^{\frac{p-2}{2}} \xi,
    \end{equation}
for all $\xi\in \mathbb{R}^{n}$.

We will denote by
 \begin{align}\label{definizionet}
        \frac{1}{\overline{q}} = \frac{1}{n} \sum_{i=1}^{n} \frac{p_i+1}{p_i+2}, \qquad t: = \begin{cases}\frac{n \overline{q}}{n- \overline{q}} & \quad \text{if} \ \overline{q} <n,\\
        \text{any exponent in } \left(\left(\frac{\overline{p}^*}{p_n}\right)',\infty \right) & \quad \text{if} \  \overline{q} \ge n.
        \end{cases}
    \end{align}

Our aim is to prove the following

For the exponents $p_i$, we shall denote by $\mathbf{p} = (p_1, p_2, \ldots, p_n)$ and by
\begin{equation*}
 W^{1,\mathbf{p}}_{\mathrm{loc}}(\Omega) = \left\{ u \in W^{1,1}_{\mathrm{loc}}(\Omega) : u_{x_i} \in L^{p_i}_{\mathrm{loc}}(\Omega), \, i = 1, \ldots, n \right\},
 \end{equation*}
the corresponding anisotropic Sobolev space.\\
We denote by $\overline{p}$ the harmonic average of $p_i$ and by $\overline{p}^*$ the Sobolev conjugate exponent of $\overline{p}$, i.e.
 \begin{align}\label{definizionePi}
 \frac{1}{\overline{p}} = \frac{1}{n} \sum_{i=1}^{n} \frac{1}{p_i}, \qquad \overline{p}^* = \begin{cases}\frac{n \overline{p}}{n- \overline{p}} & \quad \text{if} \ \overline{p} <n,\\
 \text{any exponent in } [1,\infty) & \quad \text{if} \ \overline{p} \ge n.
 \end{cases}
 \end{align}
 For an exponent $1 \le s \le \infty$, we will denote by $s'$ the H\"older conjugate of $s$.\\
In the following, we will assume that the function $\omega(x) $ belongs to the Sobolev space $W^{1,{\frac{\mathbf{p}+2}{\mathbf{p}+1}}}_{\mathrm{loc}}(\Omega)$, with ${\frac{\mathbf{p}+2}{\mathbf{p}+1}} = (\frac{p_1+2}{p_1+1}, \frac{p_2+2}{p_2+1}, \ldots, \frac{p_n+2}{p_n+1})$.\\

\noindent Let us recall the following definition of local minimizer.
\begin{dfn}
{ \em A function {$u\in W_{\rm loc}^{1,\mathbf{p}}(\Omega)$} is a local minimizer of
\eqref{functional} if, for every open subset $\tilde{\Omega} \Subset \Omega$, we have
$\mathcal{F}(u;\tilde{\Omega})\le \mathcal{F}(\varphi;\tilde{\Omega})$
for all $\varphi\in \mathcal{C}_0^{\infty}(\tilde{\Omega})$.}
\end{dfn}
The energy densities $f$ that satisfy anisotropic growth condition fit into the wider context of those with $(p,q)$-growth condition as follows
$$|\xi|^p \leq f(x,\xi) \leq (1 + |\xi|^2)^{\frac{q}{2}}, \qquad1<p<q, $$
where $p= c(p_i)$ and $q= \max \{p_i : i=1, \dots, n \}.$\\
The study of regularity properties of local minimizers to functionals with non-standard growth started with the pioneering papers by Marcellini \cite{MarcelliniEs1, MarcelliniEs2}.
 It is now well understood that, in order to obtain some regularity for the local minimizers of functionals with $(p,q)$-growth, $p<q$, a restriction between the exponents $p$ and $q$ must be imposed (see the counterexamples in \cite{Giaquinta,MarcelliniEs3}). Indeed, the gap $\frac{q}{p}$ cannot differ to much from $1$ and sufficient conditions to the regularity can be expressed as
$$\frac{q}{p}\le c(n)\displaystyle \to 1\,\,\textnormal{as}\,\,\,\,n\to \infty,$$
according to \cite{MarcelliniEs1,MarcelliniEs2}.

In recent years there has been a considerable of interest in variational problems with $(p,q)$-growth conditions (see, for example, \cite{Adi,Barone,6,12,SuggeritoRefery,Ming1,Ming2,Ming3,Ming4,Ele1,Ele2,Ele3,25,Hasto,Koch,16} and references therein). It is impossible to give a complete overview on the subject, however we refer the interested reader to the recent survey \cite{surveyMar1,surveyMing}.\\
Compared to the above mentioned papers, the difference with our work lies in the fact that the behavior of our functional at \eqref{functional} depends on the components of the gradient and, although they exhibit $(p,q)$-growth, they need to be treated with appropriate arguments.\\
Actually, the regularity of local minimizers of functionals with orthotropic structure as in \eqref{functional} is a well-studied problem: some results can be found in \cite{6,5,LemmaBrasco,Carozza,17,53,russo}, for the homogeneous case \( \omega = 0 \), and for the non-homogeneous one, we refer to \cite{BraTau,CMM}.
\\
In the autonomous case, i.e.\ $a_i(x) \equiv 1$, for all $i=1,...,n$, it has been proved in \cite{Brascop12}, for the sub-quadratic case with $\omega \in L^{1+ \frac{2}{p_1}}_{\mathrm{loc}}(\Omega)$,
that any local minimizer \( u \in W^{1,\mathbf{p}}_{\text{loc}}(\Omega) \cap L^\infty_{\text{loc}}(\Omega) \) satisfies
\begin{equation}
 |u_{x_i}|^{\frac{p_i - 2}{2}} \, u_{x_i} \in W^{1,2}_{\text{loc}}(\Omega), \quad i = 1, \dots,n. \label{diffproperties}
\end{equation}
For the super-quadratic case, Brasco et al.\ \cite{BraTau} consider the following widely degenerate orthotropic functional
$$F(u; \Omega) = \sum_{i=1}^{n} \int_{\Omega} \frac{1}{p_i} \left( |u_{x_i}| - \delta_i \right)^{p_i} \, dx - \int_{\Omega} \omega \,u \, dx, $$
with $\delta_i \ge 0$, $p_i \ge 2$, $ u \in W^{1,\mathbf{p}}_{\text{loc}}(\Omega)\cap L^\infty_{\text{loc}}(\Omega)$ and
$ \omega \in W^{1,\mathbf{p'}}_{\text{loc}}(\Omega), $ $\mathbf{p'}=(p_1', \cdots, p_n')$, where $p_i'$ is the H\"older conjugate of $p_i$.
The authors proved the higher differentiability of the local minimizers, that is
\[
\left( |u_{x_i}| - \delta_i \right)^\frac{p_i}{2}_+ \frac{u_{x_i}}{|u_{x_i}|} \in W^{1,2}_{\text{loc}}(\Omega), \quad i = 1, \dots, n.
\]
In particular, for local weak solutions of the anisotropic orthotropic $p$-Laplace equation (i.e. the case $\delta_i = 0$), they get the regularity at \eqref{diffproperties}.

We will denote by
 \begin{align}\label{definizionet}
 \frac{1}{\overline{q}} = \frac{1}{n} \sum_{i=1}^{n} \frac{p_i+1}{p_i+2}, \qquad t: = \begin{cases}\frac{n \overline{q}}{n- \overline{q}} & \quad \text{if} \ \overline{q} <n,\\
 \text{any exponent in } \left(\left(\frac{\overline{p}^*}{p_n}\right)',\infty \right) & \quad \text{if} \ \overline{q} \ge n.
 \end{cases}
 \end{align}

Our aim is to prove the following

\begin{rmk}
We observe that condition \eqref{Ipotesip} in Theorem \ref{thmBfinito} ensures that $u \in L^\infty_{\mathrm{loc}}(\Omega)$. Indeed, by Lemma \ref{LemmaTroisi}, the assumption $\omega \in W_{\mathrm{loc}}^{1,\frac{\mathbf{p}+2}{\mathbf{p}+1}}(\Omega)$ implies $\omega \in L^t_{\mathrm{loc}}(\Omega)$, where $t$ was defined at \eqref{D1}. Thus, assuming that the exponents $t$ and $p_i$ satisfy the relation $p_n < \frac{\overline{p}^*}{t'}$, we obtain that local minimizers of \eqref{functional} are locally bounded in $\Omega$ (see Theorem \ref{ulimitatoCupini} below).
If $\omega \in L^\infty_{\mathrm{loc}}(\Omega)$, then this relation becomes $p_n < \overline{p}^*$, which is the usual one when dealing with bounded minimizers (see \cite{CMMjota}).
If $\overline{p} \ge n$, this condition disappears, since according to \eqref{definizionePi} we can choose $\overline{p}^*$ large enough so that $p_1+2 < \frac{\overline{p}^*}{t'}$. \\
Moreover, since $p_i \ge 2$, for every $i=1,...,n$, it holds $\overline{q} < \overline{p}$. Therefore,
 if $\overline{p}<n$, the exponent $t$ appearing in \eqref{Ipotesip} is equal to $\frac{n \overline{q}}{n-\overline{q}}$.\\
 Eventually, we note that the exponents $\overline{p}^*$ and $t$ depend on $p_i$, hence the condition $p_n < \frac{\overline{p}^*}{t'}$ is necessary, since is not satisfied for any choice of exponents $p_i$.
\end{rmk}

\section{A regularized problem}\label{PreREG}
In this section we prove a regularity result for more regular problems with respect to \eqref{functional} that will be needed in the approximation procedure. More precisely, for $\varepsilon \geq 0$ we consider
\begin{equation}\label{Eqconepsilon}
 f_\varepsilon(x,u, \xi)= \sum_{i=1}^{n} \, \dfrac{b_i(x)}{p_i} \lvert \xi_i \rvert^{p_i} + \varepsilon(1+|\xi|^\frac{p_n}{2})^2 -\overline{\omega}(x)u
\end{equation}
with
$b_i(x):\Omega \to [0,+\infty), \, i=1, \dots, n$, non-negative, Lipschitz continuous coefficients,
i.e., the following conditions hold
\begin{equation}\label{atilde}
L_i= \sup_{x,y \in \Omega, x \neq y} \frac{|b_i(x)-b_i(y)|}{|x-y|}< \infty , \quad i=1,..,n.
\end{equation}
Setting $\tilde{f}(x, \xi)= \sum_{i=1}^{n} \, b_i(x) \lvert \xi_i \rvert^{p_i},$ one can easily check that
\begin{equation}\label{A'4}
 |D_\xi \tilde{f}(x,\xi)-D_\xi \tilde{f}(y, \xi)| \leq K |x-y|\, \, \sum_{i=1}^{n} |\xi_i|^{p_i-1} \tag{A3'}
\end{equation}
\noindent for a.e.\ $x,y \in \Omega $ and every $\xi \in \mathbb{R}^{ n}$, where $K=K(L_i, p_i)$
.\\
We recall a Lipschitz regularity Theorem for the minimizers of the functional
\begin{equation}\label{functional2}
 \mathcal{F}_\varepsilon (x, Dv) =\int_\Omega f_\varepsilon(x,v, Dv) \, dx,
\end{equation}
that can be deduced by \cite[Theorem $8.9$]{giusti} in the case $p=p_n$, where $f_\varepsilon$ was defined at \eqref{Eqconepsilon}.
\begin{thm}\label{thm2Mascolo}
 Let $\tilde{f} $ satisfy \eqref{A2}, \eqref{A3} and \eqref{A'4} and $\overline{\omega} \in L^\infty(\Omega)$.
 Let $u_\varepsilon \in W^{1,p_n}_{\mathrm{loc}}(\Omega)$ be a local minimizer of \eqref{functional2}.
Then, $u_\varepsilon \in W^{1,\infty}_{\mathrm{loc}}(\Omega)$
\end{thm}

\begin{align}
 \sum_{i=1}^n \int_{B_{\rho}} |\tau_{j,h} V_{p_i} ((u_\varepsilon)_{x_i})|^2 dx & \leq \,|h|^2 \Biggl\{ \frac{c_\beta}{(t-\rho)^2} +
\frac{c(K)}{(t-\rho)}
+ c_{\beta}
 + \frac{c \varepsilon}{(t-\rho)^2} \notag\\
& \qquad + \sum_{i=1}^n \left( \int_{B_R}|\omega_{x_i}|^{\frac{p_i+2}{p_i+1}} \, dx\right)^\frac{p_i+1}{p_i+2} \Biggr\}, \notag
 \end{align}
 and so
\begin{align}
 &\sum_{i=1}^n \int_{B_{\rho}} |\tau_{j,h} V_{p_i} ((u_\varepsilon)_{x_i})|^2 dx \leq c |h|^{2} \,\left[ 1+ \sum_{i=1}^{n} \Vert \omega_{x_i} \Vert_{L^{\frac{p_i+2}{p_i+1}}(B_{R})} \right], \label{StimaLemma}
\end{align}
for a positive constant $c = c(n,p_i,\rho, R, \lambda, K, \lvert \lvert Du \rvert \rvert_{\infty})$.
Then we have that
 $$V_{p_i}((u_\varepsilon)_{x_i}) \in W^{1,2}_{\mathrm{loc}}(\Omega), \qquad \forall i=1, \dots, n,$$
 which implies in particular that $$|(u_\varepsilon)_{x_i x_j}|^2|(u_\varepsilon)_{x_i}|^{p_i-2} \in L^1_{\mathrm{loc}}(\Omega), \qquad \forall i=1, \dots, n \text{ and } \forall j=1, \dots, n. $$
\end{proof}

\section{A priori estimates}\label{Apriorisec}
The main aim of this section is to establish the following a priori estimate which is the main step in the proof of our main result.
\begin{thm}\label{AppThm}
Let $\omega \in W^{1,\frac{\mathbf{p}+2}{\mathbf{p}+1}}_{\mathrm{loc}}(\Omega)$ and let $u \in W^{1, \mathbf{p}}_{\mathrm{loc}}(\Omega) \cap L^\infty_{\mathrm{loc}}(\Omega)$ be a local minimizer of \eqref{functional} under assumptions \eqref{A2}--\eqref{A4}, with exponents $p_i\geq 2, \forall i=1,\dots,n $, such that $p_n < p_1+2$,
and with a function $g \in L^{r}_{\mathrm{loc}}{(\Omega)}$, where $r$ satisfies the condition
\eqref{Ipotesir}.
If we assume that
\begin{equation}\label{Apriori}
V_{p_i}(u_{x_i}) \in {W^{1,2}_{\mathrm{loc}}(\Omega)}, \qquad\forall i=1,\dots,n,
\end{equation}
then the following estimates
\begin{align}\label{uxistima}
 \sum_{i=1}^{n} \left( \int_{B_{R/4}} |u_{x_i}|^{p_i +2 } dx \right) & \leq c \left(1+ \sum_{i=1}^{n} \left(\Vert u_{x_i} \Vert_{L^{p_i}(B_R)} + \Vert \omega_{x_i} \Vert_{L^\frac{p_i+2}{p_i+1}(B_{R})} \right) + \Vert g \Vert_{L^r (B_{R})} \right)^\gamma
\end{align}
and
\begin{align}
 \sum_{i=1}^{n} \left( \int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2}dx \right) & \leq c \left( 1+\sum_{i=1}^{n} \left(\Vert u_{x_i} \Vert_{L^{p_i}(B_R)} + \Vert \omega_{x_i} \Vert_{L^\frac{p_i+2}{p_i+1}(B_{R})} \right)+ \Vert g \Vert_{L^r (B_{R})} \right)^\gamma
 \label{StimaTeo1}
\end{align}
hold for every pair of concentric balls $B_{R/4} \subset B_{R} \Subset \Omega$, where $c = c(n,p_i,\lambda, \Lambda,R, {\Vert u \Vert_{\infty}})$ and $\gamma= \gamma (n,p_i, {r})$ are positive constants.
\end{thm}
\begin{proof}
We start by observing that, thanks to assumption \eqref{Ipotesip}, Theorem \ref{ulimitatoCupini} and Lemma \ref{LemmaTroisi}, we have $u \in L^\infty_{\mathrm{loc}}(\Omega)$. Therefore, by virtue of the local boundedness of $u$ and of assumption \eqref{Apriori}, an application of Proposition \ref{LemmaBrasco} gives that
\begin{equation}
 u_{x_k} \in L^{p_k+2}_{\mathrm{loc}}(\Omega) \qquad \text{for every } k \in \mathbb{N}. \label{highint}
\end{equation}
Fix a ball $B_R \Subset \Omega$ and consider radii $\frac{R}{4}< \rho< s<t<t'<{\sigma}<R$, a cut-off function $\eta \in C_0^{\infty}(B_t)$, with $\eta=1$ on $B_{s}$, $0 \leq \eta \leq 1$, $|D \eta | \leq \frac{c}{t-s}$ and $|h|\leq \frac{t' - t}{2}$.
We test the Euler-Lagrange equation of \eqref{functional} with the function $$\varphi = \tau_{j,-h}(\eta^2 \tau_{j,h}u),$$
thus getting
\begin{align}\label{SommaIntera}
 0= \int_{\Omega} &\langle f_{\xi} (x+e_jh, Du(x+e_jh))- f_{\xi} (x+e_jh, Du(x)) , \eta^2 \tau_{j,h} Du \rangle \, dx \notag \\
 &+ \int_{\Omega} \langle f_{\xi} (x+e_jh, Du(x+e_jh))- f_{\xi} (x+e_jh, Du(x)) , 2 \eta D \eta \tau_{j,h} u \rangle \, dx \notag \\
 &+ \int_{\Omega} \langle f_{\xi} (x+e_jh, Du(x))- f_{\xi} (x, Du(x)) , 2 \eta D \eta \tau_{j,h} u \rangle \, dx \notag \\
 &+ \int_{\Omega} \langle f_{\xi} (x+e_jh, Du(x))- f_{\xi} (x, Du(x)) , \eta^2 \tau_{j,h} Du \rangle \, dx \notag \\
 &- \int_{\Omega} \tau_{j,h}\omega(x)\eta^2 \tau_{j,h} u(x) \, dx \notag\\
 & =: I_1 + I_2 + I_3 + I_4+I_5. \notag
\end{align}
From the previous equality, we deduce that
\begin{equation}\label{SommaInt}
 I_1 \leq |I_2| + |I_3| + |I_4|+|I_5| .
\end{equation}
By virtue of assumption \eqref{A2}, we infer
\begin{align}\label{I1}
 I_1 &= \int_{\Omega} \langle f_{\xi} (x+e_j h, Du(x+ e_jh))- f_{\xi} (x+ e_j h, Du(x)) , \eta^2 \tau_{j,h} Du \rangle \, dx \notag \\
 & \geq l \int_{\Omega} \eta^2 \sum_{i=1}^{n}\, \left( \, |u_{x_i}(x+e_jh)|^2+|u_{x_i}(x) |^2 \, \right)^\frac{p_i -2}{2}| \tau_{j,h} u_{x_i} |^2 dx =: l \textbf{\textbf{RHS}}.
\end{align}

\begin{rmk}
We observe that condition \eqref{Ipotesip} in Theorem \ref{thmBfinito} ensures that $u \in L^\infty_{\mathrm{loc}}(\Omega)$. Indeed, by Lemma \ref{LemmaTroisi}, the assumption $\omega \in  W_{\mathrm{loc}}^{1,\frac{\mathbf{p}+2}{\mathbf{p}+1}}(\Omega)$ implies $\omega \in L^t_{\mathrm{loc}}(\Omega)$, where $t$ was defined at \eqref{D1}. Thus, assuming that the exponents $t$ and $p_i$ satisfy the relation $p_n < \frac{\overline{p}^*}{t'}$, we obtain that local minimizers of \eqref{functional} are locally bounded in $\Omega$ (see Theorem \ref{ulimitatoCupini} below).
If $\omega \in L^\infty_{\mathrm{loc}}(\Omega)$, then this relation becomes $p_n  < \overline{p}^*$, which is the usual one when dealing with bounded minimizers (see \cite{CMMjota}).
If $\overline{p} \ge n$, this condition disappears, since according to \eqref{definizionePi} we can choose $\overline{p}^*$ large enough so that $p_1+2 < \frac{\overline{p}^*}{t'}$. \\
Moreover, since $p_i \ge 2$, for every $i=1,...,n$, it holds $\overline{q} < \overline{p}$. Therefore,
 if $\overline{p}<n$,  the exponent $t$ appearing in \eqref{Ipotesip} is equal to $\frac{n \overline{q}}{n-\overline{q}}$.\\
 Eventually, we note that the exponents $\overline{p}^*$ and $t$ depend on $p_i$, hence the condition $p_n < \frac{\overline{p}^*}{t'}$ is necessary, since is not satisfied for any choice of exponents $p_i$.
\end{rmk}

Now, we briefly describe the strategy of the proof, which, as usual, will consist of an approximation argument, a uniform a priori estimate, and a passage to the limit. It is well known that the a priori estimate needs to be established for sufficiently regular minimizers. Therefore, the first step is to establish higher differentiability for minimizers of functionals with Lipschitz continuous coefficients, since this result is not available in the literature. As usual the higher differentiability is achieved with the use of the well known difference quotient methods that is frequently used in this context.\\
More specifically, for a given $\varepsilon \geq 0$, we investigate the function
\[
f_\varepsilon(x,u, \xi) = \sum_{i=1}^{n} \dfrac{b_i(x)}{p_i} |\xi_i|^{p_i} + \varepsilon \left( 1 + |\xi|^{\frac{p_n}{2}} \right)^2-\omega(x)u
\]
where the coefficients $b_i(x): \Omega \to \mathbb{R}$ are non-negative and satisfy the following condition
\[
L_i = \sup_{x, y \in \Omega, x \neq y} \frac{|b_i(x) - b_i(y)|}{|x - y|} < \infty.
\]
The first goal is to analyze the regularity properties of minimizers of the following functional
\[
\mathcal{F}_\varepsilon(u, \Omega) = \int_\Omega f_\varepsilon(x,u, Du) \, dx.
\]
After, we aim to establish an uniform a priori estimate, which plays a crucial role in the proof of our main result. Finally, Theorem \ref{thmBfinito} follows by proving that the a priori estimate is preserved in passing to the limit.\\

The plan of the paper is briefly described. After summarizing some known results and fixing a few notations, we prove a lemma for approximating functionals to be used in the a priori estimates in Section \ref{PreREG}. Section \ref{Apriorisec} is devoted to the proof of some a priori estimates for solutions to a family of approximating problems. Next, in Section \ref{Appro}, we pass to the limit in the approximating problems.

To prove Theorem~\ref{thmBfinito} the strategy is described as follows: it “will consist of an approximation argument, a uniform a priori estimate, and a passage to the limit.” Since “the a priori estimate needs to be established for sufficiently regular minimizers,” the first step is “to establish higher differentiability for minimizers of functionals with Lipschitz continuous coefficients,” a result the authors note “is not available in the literature.” This higher differentiability is obtained “with the use of the well known difference quotient methods.”

More specifically, for “a given $\varepsilon \ge 0$,” they introduce the approximating integrand
\[
f_\varepsilon(x,u,\xi)=\sum_{i=1}^n \frac{b_i(x)}{p_i}|\xi_i|^{p_i}+\varepsilon\bigl(1+|\xi|^{\frac{p_n}{2}}\bigr)^2-\omega(x)u,
\]
with nonnegative coefficients $b_i$ that are Lipschitz, i.e. $\sup_{x\ne y}\frac{|b_i(x)-b_i(y)|}{|x-y|}<\infty$, and the functional
\[
\mathcal F_\varepsilon(u,\Omega)=\int_\Omega f_\varepsilon(x,u,Du)\,dx.
\]
The “first goal is to analyze the regularity properties of minimizers” of $\mathcal F_\varepsilon$. After that, they “aim to establish an uniform a priori estimate,” and finally Theorem~\ref{thmBfinito} is obtained by “proving that the a priori estimate is preserved in passing to the limit.”

To prove the main theorem, the strategy is described as follows: it “will consist of an approximation argument, a uniform a priori estimate, and a passage to the limit.” Since “the a priori estimate needs to be established for sufficiently regular minimizers,” the first step is “to establish higher differentiability for minimizers of functionals with Lipschitz continuous coefficients,” a result the authors note “is not available in the literature.” This higher differentiability is obtained “with the use of the well known difference quotient methods.”

More specifically, for “a given $\varepsilon \ge 0$,” they introduce the approximating integrand
\[
f_\varepsilon(x,u,\xi)=\sum_{i=1}^n \frac{b_i(x)}{p_i}|\xi_i|^{p_i}+\varepsilon\bigl(1+|\xi|^{\frac{p_n}{2}}\bigr)^2-\omega(x)u,
\]
with nonnegative coefficients $b_i$ that are Lipschitz, i.e. $\sup_{x\ne y}\frac{|b_i(x)-b_i(y)|}{|x-y|}<\infty$, and the functional
\[
\mathcal F_\varepsilon(u,\Omega)=\int_\Omega f_\varepsilon(x,u,Du)\,dx.
\]
The “first goal is to analyze the regularity properties of minimizers” of $\mathcal F_\varepsilon$. After that, they “aim to establish an uniform a priori estimate,” and finally the main theorem is obtained by “proving that the a priori estimate is preserved in passing to the limit.”,

\label{thmBfinito}
Let $\omega \in  W_{\mathrm{loc}}^{1,\frac{\mathbf{p}+2}{\mathbf{p}+1}}(\Omega) $ and let $u \in {W_{\mathrm{loc}}^{1,\mathbf{p}}}(\Omega)$ be a local minimizer of
\begin{equation}\label{functional}
 \mathcal{F}(u,\Omega):= \, \int_\Omega \left[f(x, Du)-\omega(x)u(x) \right] dx,
\end{equation}
under assumptions
\begin{equation}\label{A2}
    \langle D_\xi f(x, \xi)-D_\xi f(x, \eta),\xi-\eta\rangle  \geq  l \sum_{i=1}^{n}(|\xi_i|^2+|\eta_i |^2)^\frac{p_i -2}{2}|\xi_i - \eta_i |^2 \tag{A1}
\end{equation}
and
\begin{equation}\label{A4}
    |D_\xi f(x,\xi)-D_\xi f(y, \xi)| \leq |x-y|\,(g(x)+g(y)) \, \sum_{i=1}^{n} |\xi_i|^{p_i-1} \tag{A3}
\end{equation}
with exponents $p_i\geq 2, \forall i=1,\dots,n $, such that
\begin{equation}\label{Ipotesip}
p_n <
\begin{cases}
    \min \Big\{ \dfrac{\overline{p}^*}{t'} ,p_1 +2 \Big\} \, &\text{if } \overline{p} < n,  \\
     p_1 +2 \quad &\text{otherwise},
    \end{cases}
\end{equation}
and with a function $g \in {L^{r}_{\mathrm{loc}}(\Omega)}$, where $r$ satisfies the condition
\begin{equation}\label{Ipotesir}
   r> p_n + 2 .
\end{equation}
Then,
\begin{equation*}
{V_{p_i}(u_{x_i}) \in W^{1,2}_{\mathrm{loc}}(\Omega)} \qquad\forall i=1,\dots,n,
\end{equation*}
and the following estimates
\begin{align*}
    &\sum_{i=1}^{n} \left(  \int_{B_{R/4}} |u_{x_i}|^{p_i +2 } dx \right) \notag\\
    &\leq c  \left( 1+\sum_{i=1}^{n} \left(\Vert u_{x_i} \Vert_{L^{p_i}(B_R)} + {\Vert \omega_{x_i} \Vert_{L^\frac{p_i+2}{p_i+1}(B_{R})}} \right) +\Vert \omega\Vert_{L^t(B_R)}+ \Vert g \Vert_{L^r (B_{R})} \right)^\sigma
\end{align*}
and
\begin{align*}
        &\sum_{i=1}^{n}  \left( \int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2}dx \right) \notag\\
         &\leq c  \left( 1+\sum_{i=1}^{n} \left(\Vert u_{x_i} \Vert_{L^{p_i}(B_R)} + {\Vert \omega_{x_i} \Vert_{L^\frac{p_i+2}{p_i+1}(B_{R})}} \right) +\Vert \omega\Vert_{L^t(B_R)}+ \Vert g \Vert_{L^r (B_{R})} \right)^\sigma
\end{align*}
hold for every pair of concentric balls $B_{R/4} \subset B_{R} \Subset \Omega$, where $c = c(n,p_i,\lambda, \Lambda,R, \Vert u \Vert_{\infty})$ and $\sigma= \sigma (n,p_i, r)$ are positive constants.,

Question:
Let \(\Omega\subset \mathbb{R}^n\), let \(\mathbf p=(p_1,\dots,p_n)\) with \(p_i\ge 2\) for all \(i\), and define the anisotropic Sobolev space
\[
W^{1,\mathbf p}_{\mathrm{loc}}(\Omega)=\{u\in W^{1,1}_{\mathrm{loc}}(\Omega): u_{x_i}\in L^{p_i}_{\mathrm{loc}}(\Omega)\ \text{for }i=1,\dots,n\}.
\]
Let
\[
\frac1{\overline p}=\frac1n\sum_{i=1}^n \frac1{p_i},
\qquad
\overline p^*=\frac{n\overline p}{n-\overline p}\ \text{if }\overline p<n,
\]
and let \(t'\) denote the H\"older conjugate of \(t\). Assume \(\omega\in L^t_{\mathrm{loc}}(\Omega)\) and \(\omega\in W^{1,\frac{\mathbf p+2}{\mathbf p+1}}_{\mathrm{loc}}(\Omega)\), meaning \(\omega_{x_i}\in L^{\frac{p_i+2}{p_i+1}}_{\mathrm{loc}}(\Omega)\) for each \(i\). Let \(g\in L^r_{\mathrm{loc}}(\Omega)\) with \(r>p_n+2\), and let \(u\in W^{1,\mathbf p}_{\mathrm{loc}}(\Omega)\) be a local minimizer of
\[
\mathcal F(u;\Omega)=\int_\Omega \bigl[f(x,Du)-\omega(x)u(x)\bigr]\,dx.
\]
Assume furthermore that for all \(x,y\in\Omega\) and \(\xi,\eta\in\mathbb R^n\),
\[
\langle D_\xi f(x,\xi)-D_\xi f(x,\eta),\xi-\eta\rangle
\ge l\sum_{i=1}^n (|\xi_i|^2+|\eta_i|^2)^{\frac{p_i-2}{2}}|\xi_i-\eta_i|^2,
\]
and
\[
|D_\xi f(x,\xi)-D_\xi f(y,\xi)|
\le |x-y|\,(g(x)+g(y))\sum_{i=1}^n |\xi_i|^{p_i-1}.
\]
Assume also that
\[
p_n<
\begin{cases}
\min\left\{\dfrac{\overline p^*}{t'},\,p_1+2\right\}, & \overline p<n,\\[4pt]
p_1+2, & \overline p\ge n.
\end{cases}
\]
For \(s\in\mathbb R\), write \(V_{p_i}(s)=|s|^{\frac{p_i-2}{2}}s\). Under these assumptions, which quantitative estimate holds?

For every \(i=1,\dots,n\),
\[
V_{p_i}(u_{x_i})\in W^{1,2}_{\mathrm{loc}}(\Omega).
\]
Moreover, for every pair of concentric balls \(B_{R/4}\subset B_R\Subset \Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma,
\]
and, for each fixed \(j=1,\dots,n\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i x_j}|^2\,|u_{x_i}|^{p_i-2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma,
\]
where \(c>0\) depends on \(n\), the exponents \(p_i\), the structural constants in the hypotheses, \(R\), and \(\|u\|_\infty\), while \(\sigma>0\) depends only on \(n\), the exponents \(p_i\), and \(r\).

Choices:
B.
For every \(i=1,\dots,n\),
\[
V_{p_i}(u_{x_i})\in W^{1,2}_{\mathrm{loc}}(\Omega).
\]
Moreover, for every pair of concentric balls \(B_{R/4}\subset B_R\Subset \Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^{p_n+2}(B_R)}\right)^\sigma,
\]
and, for each fixed \(j=1,\dots,n\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i x_j}|^2\,|u_{x_i}|^{p_i-2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^{p_n+2}(B_R)}\right)^\sigma,
\]
where \(c>0\) depends on \(n\), the exponents \(p_i\), the structural constants in the hypotheses, \(R\), and \(\|u\|_\infty\), while \(\sigma>0\) depends only on \(n\), the exponents \(p_i\), and \(r\).
C.
For every pair of concentric balls \(B_{R/4}\subset B_R\Subset \Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma,
\]
where \(c>0\) depends on \(n\), the exponents \(p_i\), the structural constants in the hypotheses, \(R\), and \(\|u\|_\infty\), while \(\sigma>0\) depends only on \(n\), the exponents \(p_i\), and \(r\).
D.
For every \(i=1,\dots,n\),
\[
V_{p_i}(u_{x_i})\in W^{1,2}_{\mathrm{loc}}(\Omega).
\]
Moreover, there exist constants \(c>0\) and \(\sigma>0\), depending only on \(n\), the exponents \(p_i\), the structural constants in the hypotheses, and \(r\), such that for every pair of concentric balls \(B_{R/4}\subset B_R\Subset \Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma,
\]
and, for each fixed \(j=1,\dots,n\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i x_j}|^2\,|u_{x_i}|^{p_i-2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma.
\]
E.
For every \(i=1,\dots,n\),
\[
V_{p_i}(u_{x_i})\in W^{1,2}_{\mathrm{loc}}(\Omega).
\]
Moreover, for every pair of concentric balls \(B_{R/4}\subset B_R\Subset \Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma,
\]
and, for every \(j=1,\dots,n\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i x_j}|^2\,|u_{x_i}|^{p_i}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma,
\]
where \(c>0\) depends on \(n\), the exponents \(p_i\), the structural constants in the hypotheses, \(R\), and \(\|u\|_\infty\), while \(\sigma>0\) depends only on \(n\), the exponents \(p_i\), and \(r\).

dropped the higher-differentiability conclusion and second estimate

dependence of constants on R and \|u\|_\infty

weighted second-derivative structure coming from V_{p_i}

The stem does not state or strongly hint at the correct option. It gives hypotheses and asks for the resulting conclusion, but the answer is not explicitly embedded in the wording.

This is essentially a theorem-recall item: the stem lists the full hypotheses of a regularity theorem and asks which conclusion holds. That makes it very close to a direct restatement rather than an independently structured problem.

There is some reasoning required to distinguish the exact theorem conclusion from nearby variants (endpoint integrability, weaker true claim, overly strong uniformity, unweighted Hessian control). However, the item mainly tests precise recall/recognition of the theorem rather than substantial generative mathematical reasoning.

The distractors are strong: one is a weaker true statement, others reflect common failure modes such as endpoint exponent mistakes, unjustified uniformity of constants, and confusing weighted second-derivative estimates with full Hessian L^2 regularity. They are plausible and meaningfully distinct.

A technically well-constructed but theorem-recall-heavy MCQ. It avoids answer leakage and has high-quality distractors, yet it is largely tautological and only moderately tests reasoning.


Question:
Let \(\Omega\subset \mathbb{R}^n\), let \(\mathbf p=(p_1,\dots,p_n)\) with \(p_i\ge 2\) for all \(i\), and define the anisotropic Sobolev space
\[
W^{1,\mathbf p}_{\mathrm{loc}}(\Omega)=\{u\in W^{1,1}_{\mathrm{loc}}(\Omega): u_{x_i}\in L^{p_i}_{\mathrm{loc}}(\Omega)\ \text{for }i=1,\dots,n\}.
\]
Let
\[
\frac1{\overline p}=\frac1n\sum_{i=1}^n \frac1{p_i},
\qquad
\overline p^*=\frac{n\overline p}{n-\overline p}\ \text{if }\overline p<n,
\]
and let \(t'\) denote the H\"older conjugate of \(t\). Assume \(\omega\in L^t_{\mathrm{loc}}(\Omega)\) and \(\omega\in W^{1,\frac{\mathbf p+2}{\mathbf p+1}}_{\mathrm{loc}}(\Omega)\), meaning \(\omega_{x_i}\in L^{\frac{p_i+2}{p_i+1}}_{\mathrm{loc}}(\Omega)\) for each \(i\). Let \(g\in L^r_{\mathrm{loc}}(\Omega)\) with \(r>p_n+2\), and let \(u\in W^{1,\mathbf p}_{\mathrm{loc}}(\Omega)\) be a local minimizer of
\[
\mathcal F(u;\Omega)=\int_\Omega \bigl[f(x,Du)-\omega(x)u(x)\bigr]\,dx.
\]
Assume furthermore that for all \(x,y\in\Omega\) and \(\xi,\eta\in\mathbb R^n\),
\[
\langle D_\xi f(x,\xi)-D_\xi f(x,\eta),\xi-\eta\rangle
\ge l\sum_{i=1}^n (|\xi_i|^2+|\eta_i|^2)^{\frac{p_i-2}{2}}|\xi_i-\eta_i|^2,
\]
and
\[
|D_\xi f(x,\xi)-D_\xi f(y,\xi)|
\le |x-y|\,(g(x)+g(y))\sum_{i=1}^n |\xi_i|^{p_i-1}.
\]
Assume also that
\[
p_n<
\begin{cases}
\min\left\{\dfrac{\overline p^*}{t'},\,p_1+2\right\}, & \overline p<n,\\[4pt]
p_1+2, & \overline p\ge n.
\end{cases}
\]
For \(s\in\mathbb R\), write \(V_{p_i}(s)=|s|^{\frac{p_i-2}{2}}s\). Under these assumptions, which quantitative estimate holds?

Choices:
B.
For every \(i=1,\dots,n\),
\[
V_{p_i}(u_{x_i})\in W^{1,2}_{\mathrm{loc}}(\Omega).
\]
Moreover, for every pair of concentric balls \(B_{R/4}\subset B_R\Subset \Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^{p_n+2}(B_R)}\right)^\sigma,
\]
and, for each fixed \(j=1,\dots,n\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i x_j}|^2\,|u_{x_i}|^{p_i-2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^{p_n+2}(B_R)}\right)^\sigma,
\]
where \(c>0\) depends on \(n\), the exponents \(p_i\), the structural constants in the hypotheses, \(R\), and \(\|u\|_\infty\), while \(\sigma>0\) depends only on \(n\), the exponents \(p_i\), and \(r\).
C.
For every pair of concentric balls \(B_{R/4}\subset B_R\Subset \Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma,
\]
where \(c>0\) depends on \(n\), the exponents \(p_i\), the structural constants in the hypotheses, \(R\), and \(\|u\|_\infty\), while \(\sigma>0\) depends only on \(n\), the exponents \(p_i\), and \(r\).
D.
For every \(i=1,\dots,n\),
\[
V_{p_i}(u_{x_i})\in W^{1,2}_{\mathrm{loc}}(\Omega).
\]
Moreover, there exist constants \(c>0\) and \(\sigma>0\), depending only on \(n\), the exponents \(p_i\), the structural constants in the hypotheses, and \(r\), such that for every pair of concentric balls \(B_{R/4}\subset B_R\Subset \Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma,
\]
and, for each fixed \(j=1,\dots,n\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i x_j}|^2\,|u_{x_i}|^{p_i-2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma.
\]
E.
For every \(i=1,\dots,n\),
\[
V_{p_i}(u_{x_i})\in W^{1,2}_{\mathrm{loc}}(\Omega).
\]
Moreover, for every pair of concentric balls \(B_{R/4}\subset B_R\Subset \Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma,
\]
and, for every \(j=1,\dots,n\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i x_j}|^2\,|u_{x_i}|^{p_i}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma,
\]
where \(c>0\) depends on \(n\), the exponents \(p_i\), the structural constants in the hypotheses, \(R\), and \(\|u\|_\infty\), while \(\sigma>0\) depends only on \(n\), the exponents \(p_i\), and \(r\).

The stem gives only hypotheses and notation; it does not explicitly reveal the conclusion. The correct estimate is not stated or strongly telegraphed in the prompt itself.

This is essentially a direct theorem-recall item: after listing the full assumptions, it asks which estimate holds. The correct option is the theorem’s conclusion almost verbatim rather than a new inferential task.

There is some reasoning pressure because the options differ in subtle but meaningful ways (integrability exponent for g, dependence of constants, exact weighted second-derivative term, weaker true conclusion). However, success depends more on recognizing the precise theorem statement than on generating a conclusion from first principles.

The distractors are mathematically plausible and target realistic failure modes: weakening the conclusion, altering the threshold on g, misstating parameter dependence, and changing the natural V_p-based weight from p_i-2 to p_i.

A mathematically careful item with strong distractors, but it is largely a verbatim theorem-identification question rather than a genuinely generative reasoning task.