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Wolff potential estimates for elliptic obstacle problems with generalized Orlicz growth

Qi Xiong, Xing Fu

TL;DR

This work addresses elliptic obstacle problems with generalized Orlicz growth, capturing variable-exponent and double-phase behaviors under measure data. By developing a Musielak–Orlicz framework and employing Wolff potentials, the authors obtain pointwise gradient and oscillation estimates that lead to $C^{1,α}$ regularity, while avoiding higher obstacle regularity assumptions. The results extend Calderón–Zygmund theory to obstacle problems in this broad setting, including Reifenberg-flat domains and weighted Orlicz spaces, and provide a unified treatment across all $p$-regimes. The findings offer sharp quantitative control of gradients via Wolff potentials, with potential implications for PDE models with nonstandard growth and measure data.

Abstract

This paper investigates elliptic obstacle problems with generalized Orlicz growth involving measure data, which includes Orlicz growth, variable exponent growth, and double-phase growth as specific cases of this setting. First, we establish the existence of solutions in the Musielak-Orlicz space. Then, we derive pointwise and oscillation gradient estimates for solutions in terms of the non-linear Wolff potentials, assuming minimal conditions on the obstacle. These estimates subsequently lead to $C^{1,α}$-regularity results for the solutions.

Wolff potential estimates for elliptic obstacle problems with generalized Orlicz growth

TL;DR

This work addresses elliptic obstacle problems with generalized Orlicz growth, capturing variable-exponent and double-phase behaviors under measure data. By developing a Musielak–Orlicz framework and employing Wolff potentials, the authors obtain pointwise gradient and oscillation estimates that lead to regularity, while avoiding higher obstacle regularity assumptions. The results extend Calderón–Zygmund theory to obstacle problems in this broad setting, including Reifenberg-flat domains and weighted Orlicz spaces, and provide a unified treatment across all -regimes. The findings offer sharp quantitative control of gradients via Wolff potentials, with potential implications for PDE models with nonstandard growth and measure data.

Abstract

This paper investigates elliptic obstacle problems with generalized Orlicz growth involving measure data, which includes Orlicz growth, variable exponent growth, and double-phase growth as specific cases of this setting. First, we establish the existence of solutions in the Musielak-Orlicz space. Then, we derive pointwise and oscillation gradient estimates for solutions in terms of the non-linear Wolff potentials, assuming minimal conditions on the obstacle. These estimates subsequently lead to -regularity results for the solutions.

Paper Structure

This paper contains 6 sections, 23 theorems, 251 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Existence result
  4. Calderón--Zygmund estimates
  5. Comparison estimates
  6. The proof of gradient estimates

Key Result

Theorem 1.13

Assume that $G \in \Phi_{\rm{c}}(\Omega)$ satisfies ainc, adec, $1<p \leqslant q$, azero and aone conditions, $\mu \in \mathcal{M}_{b}(\Omega)$ and $\psi \in W^{1,G}(\Omega)$. Let $h \in W^{1,G}(\Omega)$ be given boundary data with $h\geqslant \psi$ a.e. in $\Omega$ and $u_i \in h+W_{0}^{1,G}(\Ome for all $v \in h+W_{0}^{1,G}(\Omega)$ with $v \geqslant \psi$ a.e. in $\Omega$, where $f_i \in L^1(

Theorems & Definitions (59)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • Remark 1.6
  • Definition 1.7
  • Remark 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 49 more