Regularity and uniqueness to multi-phase problem with variable exponent

TL;DR

This paper introduces a new class of multi-phase operators with variable exponents, exploring their regularity, existence, and uniqueness in inhomogeneous materials, extending classical Sobolev results to Musielak-Orlicz spaces.

Contribution

It develops the theory of multi-phase variable exponent operators, establishing their regularity, boundedness, and monotonicity, and proves existence and uniqueness of solutions for related Dirichlet problems.

Findings

Operators are bounded, continuous, and strictly monotone.

Existence of nontrivial solutions under general conditions.

Local regularity results for minimizers of associated functionals.

Abstract

In this paper, we consider a new class of multi phase operators with variable exponents, which reflects the inhomogeneous characteristics of hardness changes when multiple different materials are combined together. We at first deal with the corresponding functional spaces, namely the Musielak-Orlicz Sobolev spaces, hence we investigate their regularity properties and the extension of the classical Sobolev embedding results to the new context. Then, we focus on the regularity properties of our operators, and prove that these operators are bounded, continuous, strictly monotone, coercive and satisfy the (S_+)-property. Further, we discuss suitable problems driven by such operators. In particular, we deal with Dirichlet problems in which the nonlinearity is gradient dependent. Under very general assumptions, we establish the existence of a nontrivial solution for such problems. Also, we give additional conditions on the nonlinearity which guarantee the uniqueness of solution. Lastly, we produce some local regularity results (namely, Caccioppoli-type inequality, Sobolev-Poincar\'e-type inequalities and higher integrability) for minimizers of the integral functionals corresponding to the operators.