Identification of discontinuous parameters in double phase obstacle problems

Abstract

In this article, we investigate the inverse problem of identification of a discontinuous parameter and a discontinuous boundary datum to an elliptic inclusion problem involving a double phase differential operator, a multivalued convection term (a multivalued reaction term depending on the gradient), a multivalued boundary condition and an obstacle constraint. First, we apply a surjectivity theorem for multivalued mappings, which is formulated by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone mapping to examine the existence of a nontrivial solution to the double phase obstacle problem, which exactly relies on the first eigenvalue of the Steklov eigenvalue problem for the pp-Laplacian. Then, a nonlinear inverse problem driven by the double phase obstacle equation is considered. Finally, by introducing the parameter-to-solution-map, we establish a continuous result of Kuratowski type and prove the solvability of the inverse problem.

Keywords

• discontinuous parameter
• double phase operator
• elliptic obstacle problem
• inverse problem
• mixed boundary condition
• multivalued convection
• steklov eigenvalue problem
• 35j20
• 35j25
• 35j60
• 35r30
• 49n45
• 65j20
• – nonlinear analysis: perspectives and synergies