Double Phase Implicit Obstacle Problems with Convection and Multivalued Mixed Boundary Value Conditions
Abstract
In this paper we consider a mixed boundary value problem with a nonhomogeneous, nonlinear differential operator (called a double phase operator), a nonlinear convection term (a reaction term depending on the gradient), three multivalued terms, and an implicit obstacle constraint. Under very general assumptions on the data, we prove that the solution set of such an implicit obstacle problem is nonempty (so there is at least one solution) and weakly compact. The proof of our main result uses the Kakutani-Ky Fan fixed point theorem for multivalued operators along with the theory of nonsmooth analysis and variational methods for pseudomonotone operators.
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SIAM Journal on Mathematical Analysis
Pages: 1898 - 1926
DOI: 10.1137/21M1441195
ISSN (online): 1095-7154
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© 2022, Society for Industrial and Applied Mathematics.
History
Submitted: 17 August 2021
Accepted: 1 November 2021
Published online: 31 March 2022
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Funding Information
National Science Center of Poland : 2017/25/N/ST1/00611
Yulin Normal University : G2020ZK07
Romanian Ministry of Research, Innovation, and Digitization : PCE 137/202
H2020 Marie Skłodowska-Curie Actions https://doi.org/10.13039/100010665 : 823731 CONMECH
National Natural Science Foundation of China https://doi.org/10.13039/501100001809 : 12001478, 12026255, 12026256
Natural Science Foundation of Guangxi Province https://doi.org/10.13039/501100004607 : 2020GXNSFBA297137
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