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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Existence and regularity of minimizers for some spectral functionals with perimeter constraint
TL;DR: In this article, it was shown that the shape optimization problem has a solution for any k-in-N$ and dimension $d, and that every solution is a bounded connected open set with boundary which is a closed set of Hausdorff dimensions of dimension n-8.
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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.
40
Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity
TL;DR: In this paper, the authors present some open problems and obtain partial results for spectral optimization problems involving measure, torsional rigidity and rst Dirichlet eigenvalue.
Spectral Optimization Problems for Potentials and Measures
TL;DR: It is proved that the existence of global solutions in R d and that the optimal potentials or measures are equal to +1 outside a compact set.
Sharp estimates for the anisotropic torsional rigidity and the principal frequency.
TL;DR: In this paper, the authors generalize some classical estimates involving the torsional rigidity and the principal frequency of a convex domain to a class of functionals related to some anisotropic nonlinear operators.
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References
•Book
Topology Optimization: Theory, Methods, and Applications
Martin P. Bendsøe,Ole Sigmund +1 more
- 17 Sep 2011
TL;DR: In this article, the authors proposed a topology optimization by distribution of isotropic material for truss structures with anisotropic materials, based on the topology design of truss structure.
6.3K
•Book
Functions of Bounded Variation and Free Discontinuity Problems
Luigi Ambrosio,Diego Pallara,Nicola Fusco +2 more
- 01 Jan 2000
TL;DR: The Mumford-Shah functional minimiser of free continuity problems as mentioned in this paper is a special function of the Mumfordshah functional and has been shown to be a function of free discontinuity set.
5.3K
•Book
Convex analysis and variational problems
Ivar Ekeland,Roger Téman +1 more
- 01 Jan 1976
TL;DR: In this article, the authors consider non-convex variational problems with a priori estimate in convex programming and show that they can be solved by the minimax theorem.
•Book
Gradient Flows: In Metric Spaces and in the Space of Probability Measures
Luigi Ambrosio,Nicola Gigli,Giuseppe Savaré +2 more
- 01 Jan 2005
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.
4.1K
•Book
Minimal surfaces and functions of bounded variation
Enrico Giusti
- 01 Jan 1977
TL;DR: In this article, a priori estimation of the gradient of the Bernstein problem is given. But the gradient is not a priorimate of the radius of the singular set, and it is not known whether the gradient can be estimated by direct methods.
2.8K