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Musielak-Orlicz-Sobolev embeddings: Necessary and Sufficient Conditions

Ankur Pandey, Nijjwal Karak

TL;DR

The paper addresses when a Musielak-Orlicz-Sobolev space $W^{1,\!\Phi(\cdot,\cdot)}(\Omega)$ embeds into a Musielak-Orlicz space $L^{\!\Psi(\cdot,\cdot)}(\Omega)$ for $\Phi(x,t)=t^{p(x)}(\log(e+t))^{q(x)}$ on domains $\Omega\subset\mathbb{R}^n$. The authors develop a two-pronged approach: first extend the variable exponents $p(\cdot),q(\cdot)$ to $\mathbb{R}^n$ using a careful extension that preserves bounds and log-Hölder/Nekvinda properties, then apply known $\mathbb{R}^n$-embedding results to obtain sufficiency on $\Omega$; second, they establish a necessary condition by proving that a domain must satisfy the log-measure density condition if the embedding holds. The key contributions include a sharp sufficiency result for bounded Lipschitz domains under $p^+<n$ and $p^++q^+\ge1$, and a corresponding necessity result linking the embedding to a quantitative geometric density property of $\Omega$, together with the extension framework for the exponents. These results connect the variable-growth Musielak-Orlicz setting with domain geometry, offering practical criteria to verify embeddings in analysis of PDEs with nonstandard growth.

Abstract

In this paper we study the necessary and sufficient conditions on domain for Musielak-Orlicz-Sobolev embedding of the space $W^{1,Φ(\cdot,\cdot)}(Ω)$ where $Φ(x,t):=t^{p(x)}{(\log(e+t))}^{q(x)}.$

Musielak-Orlicz-Sobolev embeddings: Necessary and Sufficient Conditions

TL;DR

The paper addresses when a Musielak-Orlicz-Sobolev space embeds into a Musielak-Orlicz space for on domains . The authors develop a two-pronged approach: first extend the variable exponents to using a careful extension that preserves bounds and log-Hölder/Nekvinda properties, then apply known -embedding results to obtain sufficiency on ; second, they establish a necessary condition by proving that a domain must satisfy the log-measure density condition if the embedding holds. The key contributions include a sharp sufficiency result for bounded Lipschitz domains under and , and a corresponding necessity result linking the embedding to a quantitative geometric density property of , together with the extension framework for the exponents. These results connect the variable-growth Musielak-Orlicz setting with domain geometry, offering practical criteria to verify embeddings in analysis of PDEs with nonstandard growth.

Abstract

In this paper we study the necessary and sufficient conditions on domain for Musielak-Orlicz-Sobolev embedding of the space where
Paper Structure (3 sections, 11 theorems, 65 equations)

This paper contains 3 sections, 11 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminary Results
  3. Main Results

Key Result

Theorem 1.1

Let p satisfy (p1), (p2) and q satisfy (q1), (q2). If $p^+<n$ then for every $u\in W^{1,\Phi(\cdot,\cdot)}_0 (\Omega),$ where $\Phi(x,t):=t^{p(x)}{(\log(e+t))}^{q(x)}$ and $\Psi(x,t):=t^{p^*(x)}{(\log(e+t))}^{q(x)p^*(x)/p(x)}.$ Here $p^*(x)$ denotes the Sobolev conjugate of $p(x),$ that is $1/p^*(x) =1/p(x)- 1/n$.

Theorems & Definitions (30)

  • Theorem 1.1: HMOS10
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 20 more