Differential inclusion systems with double phase competing operators, convection, and mixed boundary conditions
Jinxia Cen, Salvatore A. Marano, Shengda Zeng
TL;DR
This work addresses existence of generalized and strong generalized solutions for differential inclusion systems with double-phase competing operators and convection under mixed boundary conditions. It combines Galerkin discretization with a surjectivity theorem for convex-compact multifunctions in Musielak–Orlicz–Sobolev spaces to obtain solutions, under a set of growth, measurability, and coercivity assumptions. The results include both generalized and strong generalized solutions, with a reduction to a weak solution when the competing effects vanish (i.e., $\max\{\alpha,\beta\}<0$). By extending the framework to double-phase, nonmonotone operators and mixed boundary data, the paper broadens applicability to multi-material and convective inclusion problems in continuum mechanics.
Abstract
In this paper, a new framework for studying the existence of generalized or strongly generalized solutions to a wide class of inclusion systems involving double-phase, possibly competing differential operators, convection, and mixed boundary conditions is introduced. The technical approach exploits Galerkin's method and a surjective theorem for multifunctions in finite dimensional spaces.