A class of elliptic mixed boundary value problems with (p, q)-Laplacian: existence, comparison and optimal control

The paper deals with two nonlinear elliptic equations with ( p ,  q )-Laplacian and the Dirichlet–Neumann–Dirichlet (DND) boundary conditions, and Dirichlet–Neumann–Neumann (DNN) boundary conditions, respectively. Under mild hypotheses, we prove the unique weak solvability of the elliptic mixed boun...

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Bibliographic Details
Published in:Zeitschrift für angewandte Mathematik und Physik 2022-08, Vol.73 (4), Article 151
Main Authors: Zeng, Shengda, Migórski, Stanisław, Tarzia, Domingo A., Zou, Lang, Nguyen, Van Thien
Format: Article
Language:English
Subjects:
Asymptotic properties
Boundary conditions
Boundary value problems
Dirichlet problem
Elliptic functions
Engineering
Mathematical Methods in Physics
Optimal control
Theoretical and Applied Mechanics
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Summary:The paper deals with two nonlinear elliptic equations with ( p ,  q )-Laplacian and the Dirichlet–Neumann–Dirichlet (DND) boundary conditions, and Dirichlet–Neumann–Neumann (DNN) boundary conditions, respectively. Under mild hypotheses, we prove the unique weak solvability of the elliptic mixed boundary value problems. Then, a comparison and a monotonicity results for the solutions of elliptic mixed boundary value problems are established. We examine a convergence result which shows that the solution of (DND) can be approached by the solution of (DNN). Moreover, two optimal control problems governed by (DND) and (DNN), respectively, are considered, and an existence result for optimal control problems is obtained. Finally, we provide a result on asymptotic behavior of optimal controls and system states, when a parameter tends to infinity.
ISSN:0044-2275
1420-9039
DOI:10.1007/s00033-022-01789-7