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Calderón-Zygmund estimates for higher order elliptic equations in Orlicz-Sobolev spaces

Julián Fernández Bonder, Pablo Ochoa, Analía Silva

TL;DR

The paper studies Calderón-Zygmund-type regularity for a fourth-order quasilinear elliptic equation in Orlicz-Sobolev spaces, proving that local data in terms of $G(f)$ transfer to higher integrability of $G(\Delta u)$. The authors develop a two-pronged approach: first establish a $q=1$ estimate and then employ an ε-regularization scheme together with maximal-function level-set techniques to obtain $G(\Delta u) \in L^q_{loc}$ for all $q\ge 1$. Key steps include constructing approximating biharmonic problems, proving convergence to a biharmonic limit, and deriving level-set decay for the Hardy–Littlewood maximal function to execute a CZ-type iteration. The results extend CZ regularity to higher-order nonlinear problems in the Orlicz setting and provide local bounds that depend on $G(u)$, $G(|\nabla u|)$, and $G(f)$, without requiring boundary regularity. This work lays groundwork for further CZ-type results for nonlinear higher-order systems, with potential applications to Lane-Emden type systems.

Abstract

In this paper we obtain Calderón-Zygmund estimates for the laplacian of the following fourth order quasilinear elliptic problem $$ Δ(g(Δu)Δu) = Δ(g(Δf)Δf). $$ where the primitive of $g(t)t$, $G(t)$, is an $N-$function. We prove that if $G(f)\in L^q$, then $G(Δu)\in L^q$ for $q\ge 1$.

Calderón-Zygmund estimates for higher order elliptic equations in Orlicz-Sobolev spaces

TL;DR

The paper studies Calderón-Zygmund-type regularity for a fourth-order quasilinear elliptic equation in Orlicz-Sobolev spaces, proving that local data in terms of transfer to higher integrability of . The authors develop a two-pronged approach: first establish a estimate and then employ an ε-regularization scheme together with maximal-function level-set techniques to obtain for all . Key steps include constructing approximating biharmonic problems, proving convergence to a biharmonic limit, and deriving level-set decay for the Hardy–Littlewood maximal function to execute a CZ-type iteration. The results extend CZ regularity to higher-order nonlinear problems in the Orlicz setting and provide local bounds that depend on , , and , without requiring boundary regularity. This work lays groundwork for further CZ-type results for nonlinear higher-order systems, with potential applications to Lane-Emden type systems.

Abstract

In this paper we obtain Calderón-Zygmund estimates for the laplacian of the following fourth order quasilinear elliptic problem where the primitive of , , is an function. We prove that if , then for .

Paper Structure

This paper contains 5 sections, 14 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Calderón-Zygmung regularity: the case $q=1$
  4. Calderón-Zygmung regularity: higher order integrability
  5. Step 3: estimates for the level sets of the Hardy-Littlewood maximal function

Key Result

Theorem 1.1

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain. Let $G(t)=\int_0^tg(s)s\,ds$ be an N-function satisfying g, G and GG-GGG, and let $u\in W_{loc}^{2, G}(\Omega)$ be a local weak solution of problem eq. Then, if Moreover, the following estimate holds: there is $C>0$ such that for any $r>0$ and $x_0\in \Omega$ so that $B_{4r}(x_0)\subset \Omega$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.6
  • Theorem 2.7
  • Lemma 2.8
  • Definition 2.9
  • ...and 19 more