Spectral optimization problems
TL;DR: In this article, the authors present a class of shape optimization problems where the cost function involves the solution of a PDE of elliptic type in the unknown domain, and focus on the existence of an optimal domain.
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Abstract: In this survey paper we present a class of shape optimization problems where the cost function involves the solution of a PDE of elliptic type in the unknown domain. In particular, we consider cost functions which depend on the spectrum of an elliptic operator and we focus on the existence of an optimal domain. The known results are presented as well as a list of still open problems. Related fields as optimal partition problems, evolution flows, Cheeger-type problems, are also considered.
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Citations
Geometrical structure of Laplacian eigenfunctions
TL;DR: In this article, the properties of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary conditions are summarized at a level accessible to scientists ranging from mathematics to physics and computer sciences.
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Geometrical structure of Laplacian eigenfunctions
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Spectral optimization for the Stekloff--Laplacian: the stability issue
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Lipschitz Regularity of the Eigenfunctions on Optimal Domains
TL;DR: In this paper, the Lipschitz regularity of the eigenfunctions of the Dirichlet Laplacian on the optimal set of spectral functionals was studied.
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Existence and Regularity Results for Some Shape Optimization Problems
Bozhidar Velichkov
- 09 Apr 2015
TL;DR: Henrot et al. as mentioned in this paper deal with the theoretical mathematical aspects of the shape optimization, concerning existence of optimal sets and their regularity, in all the practical situations above, the shape of the object in study is determined by a functional depending on the solution of a given partial differential equation.
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TL;DR: In this article, Gradient flows and curves of Maximal slopes of the Wasserstein distance along geodesics are used to measure the optimal transportation problem in the space of probability measures.
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