On renormalized solutions to elliptic inclusions with nonstandard growth

TL;DR

This paper establishes the existence and uniqueness of renormalized solutions for elliptic inclusions with nonstandard growth conditions, using Musielak–Orlicz spaces and the Minty transform to handle complex multifunctions.

Contribution

It introduces a novel approach to solving elliptic inclusions with nonstandard growth by employing Musielak–Orlicz spaces and overcoming the lack of Carathéodory selections with the Minty transform.

Findings

Proved existence and uniqueness of renormalized solutions.

Established relation between renormalized and weak solutions under additional conditions.

Extended the theory to nonpolynomial, heterogeneous, and anisotropic growth conditions.

Abstract

We study the elliptic inclusion given in the following divergence form \begin{align*} & -\mathrm{div}\, A(x,\nabla u) \ni f\quad \mathrm{in}\quad \Omega, & u=0\quad \mathrm{on}\quad \partial \Omega. \end{align*} As we assume that $f\in L^1(\Omega)$, the solutions to the above problem are understood in the renormalized sense. We also assume nonstandard, possibly nonpolynomial, heterogeneous and anisotropic growth and coercivity conditions on the maximally monotone multifunction $A$ which necessitates the use of the nonseparable and nonreflexive Musielak--Orlicz spaces. We prove the existence and uniqueness of the renormalized solution as well as, under additional assumptions on the problem data, its relation to the weak solution. The key difficulty, the lack of a Carath\'{e}odory selection of the maximally monotone multifunction is overcome with the use of the Minty transform.