Existence and Regularity of Solutions for the Elasticity System with Dirichlet-Graph Maximal Monotone Boundary Conditions

Abstract

In this work, we carried out a rigorous mathematical study of the elasticity system governed by the Lamé equations under various boundary conditions. Starting from the classical formulation with Neumann boundary conditions, we established the existence, uniqueness, and regularity of weak solutions. The second part of this work devoted to a more complex and realistic situation involving mixed boundary conditions of the Tresca–maximal monotone graph type. By reformulating the problem as a variational inequality and employing tools from monotone operator theory, we demonstrated the existence of weak solutions.