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Self-improving property for certain degenerate functionals with generalized Orlicz growth

Vertti Hietanen, Mikyoung Lee

TL;DR

The paper extends the regularity theory for variational integrals with nonstandard growth to the weighted generalized Orlicz setting. It develops a weighted framework based on Musielak–Orlicz spaces and Muckenhoupt weights, proving a Sobolev–Poincaré type inequality, a Caccioppoli inequality, and a Gehring-type self-improvement result to obtain local higher integrability of the gradient for local quasiminimizers. Additionally, existence of minimizers for the associated weighted energy is established via a direct-method approach in reflexive weighted spaces. The results generalize existing unweighted theories and accommodate special cases such as weighted variable exponent and weighted double-phase growth, with broad implications for nonlinear PDEs with nonstandard growth under weights.

Abstract

We investigate a self-improving property of variational integrals in a weighted framework under generalized Orlicz growth conditions. Assuming that the weight belongs to an appropriate Muckenhoupt class and the growth function satisfies standard structural conditions, we prove that the gradient of any local quasiminimizer has local higher integrability. In addition, we establish the existence of minimizers for the associated functional.

Self-improving property for certain degenerate functionals with generalized Orlicz growth

TL;DR

The paper extends the regularity theory for variational integrals with nonstandard growth to the weighted generalized Orlicz setting. It develops a weighted framework based on Musielak–Orlicz spaces and Muckenhoupt weights, proving a Sobolev–Poincaré type inequality, a Caccioppoli inequality, and a Gehring-type self-improvement result to obtain local higher integrability of the gradient for local quasiminimizers. Additionally, existence of minimizers for the associated weighted energy is established via a direct-method approach in reflexive weighted spaces. The results generalize existing unweighted theories and accommodate special cases such as weighted variable exponent and weighted double-phase growth, with broad implications for nonlinear PDEs with nonstandard growth under weights.

Abstract

We investigate a self-improving property of variational integrals in a weighted framework under generalized Orlicz growth conditions. Assuming that the weight belongs to an appropriate Muckenhoupt class and the growth function satisfies standard structural conditions, we prove that the gradient of any local quasiminimizer has local higher integrability. In addition, we establish the existence of minimizers for the associated functional.

Paper Structure

This paper contains 7 sections, 21 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Generalized Orlicz functions
  4. Generalized Orlicz spaces
  5. Muckenhoupt weight
  6. Existence
  7. Higher integrability

Key Result

Theorem 1.1

Let $\Omega \subset {\mathbb{R}^n}$ be a bounded domain. Let $\omega \in A_p$ and $\varphi \in {\Phi_{\mathrm{w}}}(\Omega)$ satisfy conditions aonew, ainc, and adeci with $1<p\leqslant q <\infty$. Let $u\in W^{1,\varphi}_{\omega,\mathrm{loc}}(\Omega)$ be a local quasiminimizer of Then there exists $\varepsilon>0$ depending on $n,K,p,q,\beta_0, \beta_1,L,$ and $[\omega]_{A_p}$ such that $\varphi(

Theorems & Definitions (37)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.3
  • Lemma 2.4: HHL21, Lemma 2.3
  • Lemma 2.5
  • Lemma 2.7
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • Definition 3.1
  • ...and 27 more