Inverse Problems for Double-Phase Obstacle Problems with Variable Exponents

Shengda Zeng (), Nikolaos S. Papageorgiou () and Patrick Winkert ()
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Shengda Zeng: Yulin Normal University
Nikolaos S. Papageorgiou: National Technical University
Patrick Winkert: Technische Universität Berlin

Journal of Optimization Theory and Applications, 2023, vol. 196, issue 2, No 12, 666-699

Abstract: Abstract In the present paper, we are concerned with the study of a variable exponent double-phase obstacle problem which involves a nonlinear and nonhomogeneous partial differential operator, a multivalued convection term, a general multivalued boundary condition and an obstacle constraint. Under the framework of anisotropic Musielak–Orlicz Sobolev spaces, we establish the nonemptiness, boundedness and closedness of the solution set of such problems by applying a surjectivity theorem for multivalued pseudomonotone operators and the variational characterization of the first eigenvalue of the Steklov eigenvalue problem for the p-Laplacian. In the second part, we consider a nonlinear inverse problem which is formulated by a regularized optimal control problem to identify the discontinuous parameters for the variable exponent double-phase obstacle problem. We then introduce the parameter-to-solution map, study a continuous result of Kuratowski type and prove the solvability of the inverse problem.

Keywords: Anisotropic Musielak–Orlicz Sobolev space; Discontinuous parameter; Variable exponent double-phase operator; Inverse problem; Multivalued convection; Steklov eigenvalue problem; 35J20; 35J25; 35J60; 35R30; 49N45; 65J20 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10957-022-02155-3

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