Optimal design problems governed by the nonlocal $ p $-Laplacian equation

Andrés Fuensanta; Muñoz Julio; Rosado Jesús

doi:10.3934/mcrf.2020030

Article Contents

2021, Volume 11, Issue 1: 119-141. Doi: 10.3934/mcrf.2020030

This issue Previous Article Finite-dimensional controllers for robust regulation of boundary control systems Next Article Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport

Optimal design problems governed by the nonlocal $ p $-Laplacian equation

1.
Universidad de Castilla-La Mancha, Departamento de Matemáticas and Escuela de Ingeniería Industrial y Aerospacial, Avenida Carlos Ⅲ s/n, Real Fábrica de Armas, 45071 Toledo (ESPAÑA)
2.
Universidad de Castilla-La Mancha, Departamento de Matemáticas and Facultad de CC del Medioambiente y Bioquímica, Avenida Carlos Ⅲ s/n, Real Fábrica de Armas, 45071 Toledo (ESPAÑA)Universidad de Castilla-La Mancha, Departamento de Matemáticas and Facultad de CC del Medioambiente y Bioquímica, Avenida Carlos Ⅲ s/n, Real Fábrica de Armas, 45071 Toledo (ESPAÑA)

Received: June 2019

Revised: March 2020

Early access: June 2020

Published: March 2021

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Abstract

In the present work, a nonlocal optimal design model has been considered as an approximation of the corresponding classical or local optimal design problem. The new model is driven by the nonlocal $ p $-Laplacian equation, the design is the diffusion coefficient and the cost functional belongs to a broad class of nonlocal functional integrals. The purpose of this paper is to prove the existence of an optimal design for the new model. This work is complemented by showing that the limit of the nonlocal $ p $-Laplacian state equation converges towards the corresponding local problem. Also, as in the paper by F. Andrés and J. Muñoz [J. Math. Anal. Appl. 429:288– 310], the convergence of the nonlocal optimal design problem toward the local version is studied. This task is successfully performed in two different cases: when the cost to minimize is the compliance functional, and when an additional nonlocal constraint on the design is assumed.

Keywords:

Approximation of partial differential equations,
Optimal Control,
Nonlocal elliptic equations,
G-Convergence,
p-Laplacian.

Mathematics Subject Classification: Primary:49J55;35D99;Secondary:39J92, 49J45.

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