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Higher Differentiability of Minimizers for Non-Autonomous Orthotropic Functionals

Stefania Russo

TL;DR

The paper addresses higher regularity for local minimizers of a non-autonomous anisotropic (orthotropic) functional with growth $|u_{x_i}|^{p_i}$ and coefficients $a_i(x)$ in Sobolev spaces. By imposing a dimension-dependent gap on the exponents and a Sobolev regularity $r$ on the coefficients, the authors prove that $V_{p_i}(u_{x_i})$ belongs to $W^{1,2}_{loc}$, establishing higher differentiability for each gradient component. The proof follows an approximation–a priori estimate–limit framework: first prove higher differentiability for Lipschitz-coefficient approximants, derive uniform a priori bounds, then pass to the limit to recover the result for the original functional. This advances the regularity theory for non-autonomous orthotropic functionals under non-standard growth, highlighting how coefficient regularity governs minimizer differentiability.

Abstract

We establish the higher differentiability for the minimizers of the following non-autonomous integral functionals \begin{equation*} \mathcal{F}(u,Ω):= \, \int_Ω\sum_{i=1}^{n} \, a_i(x) \lvert u_{x_i} \rvert^{p_i} dx, \end{equation*} with exponents $p_i \geq 2$ and with coefficients $a_i(x)$ that satisfy a suitable Sobolev regularity. The main result is obtained, as usual, by imposing a gap bound on the exponents $p_i$, which depends on the dimension and on the degree of regularity of the coefficients $a_i(x)$

Higher Differentiability of Minimizers for Non-Autonomous Orthotropic Functionals

TL;DR

The paper addresses higher regularity for local minimizers of a non-autonomous anisotropic (orthotropic) functional with growth and coefficients in Sobolev spaces. By imposing a dimension-dependent gap on the exponents and a Sobolev regularity on the coefficients, the authors prove that belongs to , establishing higher differentiability for each gradient component. The proof follows an approximation–a priori estimate–limit framework: first prove higher differentiability for Lipschitz-coefficient approximants, derive uniform a priori bounds, then pass to the limit to recover the result for the original functional. This advances the regularity theory for non-autonomous orthotropic functionals under non-standard growth, highlighting how coefficient regularity governs minimizer differentiability.

Abstract

We establish the higher differentiability for the minimizers of the following non-autonomous integral functionals \begin{equation*} \mathcal{F}(u,Ω):= \, \int_Ω\sum_{i=1}^{n} \, a_i(x) \lvert u_{x_i} \rvert^{p_i} dx, \end{equation*} with exponents and with coefficients that satisfy a suitable Sobolev regularity. The main result is obtained, as usual, by imposing a gap bound on the exponents , which depends on the dimension and on the degree of regularity of the coefficients

Paper Structure

This paper contains 6 sections, 15 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Difference quotient
  4. A preliminary regularity result
  5. A priori estimates
  6. Approximation

Key Result

Theorem 1.2

Let $u \in W^{1,\textbf{p}}(\Omega, \mathbb{R}^{n })$ be a local minimizer of functional under assumptions A2--A4 with exponents $p_i\geq 2, \forall i=1,\dots,n$ such that and with a function $g \in L^{r}_{\mathrm{loc}}(\Omega, \mathbb{R}^n)$, with $r$ such that Then and the following estimates and hold for every pair of concentric balls $B_{R/4} \subset B_{R} \Subset \Omega$, where $c = c(n,

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Definition 2.8
  • ...and 10 more