A class of elliptic mixed boundary value problems with (p, q)-Laplacian: existence, comparison and optimal control
A class of elliptic mixed boundary value problems with (p, q)-Laplacian: existence, comparison and optimal control
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DOI：
10.1007/s00033-022-01789-7
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发表时间：
2022-07
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通讯作者：
Shengda Zeng;S. Migórski;D. Tarzia;L. Zou;V. T. Nguyen
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作者：
Shengda Zeng;S. Migórski;D. Tarzia;L. Zou;V. T. Nguyen
本文分别讨论了两个具有(p，q)-拉普拉斯和Dirichlet-Neumann-Dirichlet(DND)边界条件和Dirichlet-Neumann-Neumann(DNN)边界条件的非线性椭圆型方程。在较弱的假设下，证明了椭圆混合边值问题唯一的弱可解性。然后，建立了椭圆型混合边值问题解的比较和单调性结果。我们检验了一个收敛结果，它表明(DND)的解可以用(DNN)的解来逼近。此外，还考虑了分别由(DND)和(DNN)支配的两个最优控制问题，得到了最优控制问题的一个存在性结果。最后，我们给出了当参数趋于无穷大时最优控制和系统状态的渐近行为的一个结果。
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.