Monotone operator methods for a class of nonlocal multi-phase variable exponent problems

TL;DR

This work addresses the existence of weak solutions for a class of nonlocal multi-phase diffusion problems with variable exponents in a Musielak–Orlicz Sobolev setting. The authors formulate the problems as operator equations driven by a Kirchhoff-type nonlocal term $\mathcal{M}(\varrho_{\mathcal{T}}(u))$ and a gradient-dependent diffusion operator, applying two monotone-operator frameworks: Browder–Minty for the first problem and pseudomonotone theory for the second with gradient dependence. Under structured assumptions on the multi-phase energy and the nonlinearity, they establish a unique nontrivial weak solution for the first problem and at least one nontrivial weak solution for the second, both in $W_0^{1,\mathcal{T}}(\Omega)$. These results extend well-posedness theory for nonlocal, variable-exponent diffusion in Musielak–Orlicz spaces and provide a methodology for heterogeneous, anisotropic diffusion modeling in composite materials.

Abstract

In this paper, we study a class of nonlocal multi-phase variable exponent problems within the framework of a newly introduced Musielak-Orlicz Sobolev space. We consider two problems, each distinguished by the type of nonlinearity it includes. To establish the existence of at least one nontrivial solution for each problem, we employ two different monotone operator methods.

Theorems & Definitions (23)

• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• Proposition 2.7
• Remark 1
• Proposition 2.8
• Definition 3.1