Logarithmic double phase embeddings with variable exponents: Necessary and Sufficient Conditions

TL;DR

This paper establishes necessary and sufficient conditions for Sobolev-type embeddings involving variable exponent and logarithmic double phase functions, characterizing domain regularity and embedding properties.

Contribution

It provides a comprehensive characterization of Sobolev embeddings with variable exponents and logarithmic phases, including domain conditions and embedding criteria.

Findings

Subcritical embedding in bounded John domains under regularity assumptions.

Embedding implies the domain satisfies the log-measure density condition.

Established equivalence between embedding properties and domain regularity.

Abstract

In this paper, we study the necessary and sufficient conditions in the domain for Sobolev-type embedding of the space $W^{1,\Phi(\cdot,\cdot)}(\Omega)$ where $\Phi(x,t):=t^{p(x)}+ a(x) t^{q(x)}\log^{r(x)}(e+t)$ with $1\leq p(x)\leq q(x).$ We have established subcritical embedding in bounded John domains under some regularity assumptions on exponents $p,$ $q,$ $r$, and $a$. Conversely, we have proved that if the embedding holds in any domain $\Omega$ in $\mathbb{R}^n,$ then $\Omega$ must satisfy the log-measure density condition.