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Inverse Problems for Double-Phase Obstacle Problems with Variable Exponents

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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.

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Acknowledgements

This project has received funding from the Natural Science Foundation of Guangxi Grant No. 2021GXNSFFA196004, the NNSF of China Grant Nos. 12001478, the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No. 823731 CONMECH, National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611, and the Startup Project of Doctor Scientific Research of Yulin Normal University No. G2020ZK07. It is also supported by the Ministry of Science and Higher Education of Republic of Poland under Grants Nos. 4004/GGPJII/H2020/2018/0 and 440328/PnH2/2019.

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Authors and Affiliations

  1. Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, 537000, Guangxi, People’s Republic of China

    Shengda Zeng

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, Jiangsu, People’s Republic of China

    Shengda Zeng

  3. Faculty of Mathematics and Computer Science, Jagiellonian University in Krakow, ul. Lojasiewicza 6, 30-348, Krakow, Poland

    Shengda Zeng

  4. Department of Mathematics, National Technical University, Zografou Campus, 15780, Athens, Greece

    Nikolaos S. Papageorgiou

  5. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Patrick Winkert

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  1. Shengda Zeng
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Zeng, S., Papageorgiou, N.S. & Winkert, P. Inverse Problems for Double-Phase Obstacle Problems with Variable Exponents. J Optim Theory Appl 196, 666–699 (2023). https://doi.org/10.1007/s10957-022-02155-3

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