Periodic evolution problems driven by Leray-Lions type operators with fully nonlinear boundary conditions

Abstract

The purpose of this work is to investigate some classes of periodic evolution problems involving Leray-Lions type operators with nonlinear boundary conditions. Two key results are presented concerning weak solutions. For cases where the source term does not depend on the solution, a general abstract method is utilized, leveraging the time-periodic condition to provide both existence and uniqueness. For scenarios involving a nonlinear source term strongly linked to the solution, the existence and uniqueness of weak solutions are achieved without requiring any sign constraints on the nonlinearity. The methodology heavily relies on Leray-Schauder topological degree, supported by innovative technical estimates.