An inverse problem for a double phase implicit obstacle problem with multivalued terms

¹ Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, Guangxi, P.R. China
² Department of Mathematics, Nanjing University, Nanjing, Jiangsu, 210093, P.R. China
³ Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30-348 Kraków, Poland
⁴ School of Science, Guangxi University of Science and Technology, Liuzhou, Guangxi 545006, China
⁵ Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059 Kraków, Poland
⁶ Department of Mathematics, University of Craiova, Street A.I. Cuza 13, 200585 Craiova, Romania
⁷ Brno University of Technology, Faculty of Electrical Engineering and Communication, Technická 3058/10, Brno 61600, Czech Republic
⁸ Simion Stoilow Institute of Mathematics of the Romanian Academy, 21 Calea Griviţei Street, 010702 Bucharest, Romania
⁹ Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany

^* Corresponding author: radulescu@inf.ucv.ro

Received: 7 October 2022
Accepted: 26 March 2023

Abstract

In this paper, we study an inverse problem of estimating three discontinuous parameters in a double phase implicit obstacle problem with multivalued terms and mixed boundary conditions which is formulated by a regularized optimal control problem. Under very general assumptions, we introduce a multivalued function called a parameter-to-solution map which admits weakly compact values. Then, by employing the Aubin-Cellina convergence theorem and the theory of nonsmooth analysis, we prove that the parameter-to-solution map is bounded and continuous in the sense of Kuratowski. Finally, a generalized regularization framework for the inverse problem is developed and a new existence theorem is provided.