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Regularity for monotone Operators and applications to homogenization of $p$-Laplace type equations

Lukas Koch, Mathias Schäffner

TL;DR

This work develops a comprehensive large-scale regularity theory for quasilinear elliptic equations with rapidly oscillating periodic coefficients of $p$-Laplacian type. By blending autonomous gradient estimates, a homogenization-corrector framework, and a large-scale Calderón–Zygmund approach, the authors obtain uniform $L^q$ gradient bounds for $q\ge p$ that are independent of the microstructure scale $\varepsilon$, and Lipschitz estimates in the nondegenerate case $\mu=1$. The homogenized operator $\overline{\bf a}$ and its correctors are shown to preserve essential structural assumptions, enabling the transfer of regularity from the homogenized problem to the oscillatory one. The results generalize classical linear and quadratic-growth theories to nonlinear, anisotropic operators, providing robust large-scale regularity and quantitative homogenization insights for $p$-Laplacian type equations with oscillating coefficients.

Abstract

In this manuscript, we provide local $L^q$-estimates for the gradient of solutions of a class of quasilinear equations whose principal part lacks strong monotonicity. These estimates are used to establish uniform large-scale $L^q$-estimates for the gradient of solutions of degenerate/singular quasilinear equations with oscillating coefficients and large-scale Lipschitz estimates for solutions of non-degenerate equations.

Regularity for monotone Operators and applications to homogenization of $p$-Laplace type equations

TL;DR

This work develops a comprehensive large-scale regularity theory for quasilinear elliptic equations with rapidly oscillating periodic coefficients of -Laplacian type. By blending autonomous gradient estimates, a homogenization-corrector framework, and a large-scale Calderón–Zygmund approach, the authors obtain uniform gradient bounds for that are independent of the microstructure scale , and Lipschitz estimates in the nondegenerate case . The homogenized operator and its correctors are shown to preserve essential structural assumptions, enabling the transfer of regularity from the homogenized problem to the oscillatory one. The results generalize classical linear and quadratic-growth theories to nonlinear, anisotropic operators, providing robust large-scale regularity and quantitative homogenization insights for -Laplacian type equations with oscillating coefficients.

Abstract

In this manuscript, we provide local -estimates for the gradient of solutions of a class of quasilinear equations whose principal part lacks strong monotonicity. These estimates are used to establish uniform large-scale -estimates for the gradient of solutions of degenerate/singular quasilinear equations with oscillating coefficients and large-scale Lipschitz estimates for solutions of non-degenerate equations.

Paper Structure

This paper contains 15 sections, 28 theorems, 348 equations.

Table of Contents

  1. Introduction and main results
  2. Main results
  3. Notation and preliminary results
  4. Function spaces
  5. -functions
  6. Energy estimates
  7. Gradient integrability for autonomous equations and proof of Theorem \ref{['thm:Lqx']}
  8. The homogenized equation and correctors
  9. Corrector and flux corrector
  10. The homogenized operator
  11. A stable assumption under homogenization
  12. Calderon-Zygmund theory on large scales
  13. Large-scale Lipschitz estimates
  14. Proof of the energy estimates Proposition \ref{['prop:regularity']} and Proposition \ref{['prop:regularity:basiclip']}
  15. Standard controlled growth conditions

Key Result

Theorem 1

Suppose that, for given $1<p<\infty$, Assumption ass:1 is satisfied. Moreover, assume that there is a nondecreasing, continuous modulus of continuity $\omega\colon [0,\infty)\to [0,\infty)$ with $\omega(0)=0$ such that Let $F\in L^{p\prime}(\Omega,\mathbb{R}^n)$ and let $u\in W^{1,p}(\Omega)$ be a weak solution to Then, for all $q\in[p,\infty)$ and every $B=B_R(x)\Subset\Omega$ there exists $C=C

Theorems & Definitions (61)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 4
  • Lemma 5
  • proof
  • Lemma 6
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 51 more