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.

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非本地控制的最佳设计问题 \ begin {document} $ p $ \ end {document}-拉普拉斯方程

在当前工作中，非局部最优设计模型已被视为对应的经典或局部最优设计问题的近似值。新模型由非局部$ p $ -Laplacian方程驱动，设计是扩散系数，成本函数属于一类广泛的非局部函数积分。本文的目的是证明新模型的最优设计的存在。通过证明非局部$ p $-拉普拉斯状态方程的极限收敛于相应的局部问题，可以对这项工作进行补充。同样，如F.Andrés和J.Muñoz的论文[J. 数学。肛门 应用 [429：288–310]，研究了非局部最优设计问题向局部版本的收敛性。在两种不同的情况下成功完成了此任务：