1,721,015 research outputs found
Minimization of the k-th eigenvalue of the Robin-Laplacian
No full textThe paper is concerned with the minimization of the k-th eigenvalue of the Laplace operator with Robin boundary conditions, among all open sets of the space satisfying a volume constraint. We prove the existence of a solution in a relaxed framework and find some qualitative properties of the optimal sets. The main idea is to see these spectral shape optimization questions as free discontinuity problems in the framework of special functions of bounded variation. One of the key difficulties (for k greater or equal than 3) comes from the fact that the eigenvalues are critical points. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM
Multiphase shape optimization problems
Full text linkThis paper is devoted to the analysis of multiphase shape optimization problems, which can formally be written as min (Formula presented.) where D ⊆ Rd is a given bounded open set, |Ωi| is the Lebesgue measure of Ωi, and m is a positive constant. For a large class of such functionals, we analyze qualitative properties of the cells and the interaction between them. Each cell is itself a subsolution for a (single-phase) shape optimization problem, from which we deduce properties like finite perimeter, inner density, separation by open sets, absence of triple junction points, etc. As main examples we consider functionals involving the eigenvalues of the Dirichlet Laplacian of each cell, i.e., Fi = λki
A surgery result for the spectrum of the Dirichlet Laplacian
No full textIn this paper we give a method to geometrically modify an open set such that the first k eigenvalues of the Dirichlet Laplacian and its perimeter are not increasing, its measure remains constant, and both perimeter and diameter decrease below a certain threshold. The key point of the analysis relies on the properties of the shape subsolutions for the torsion energy. As well, we apply this result to prove existence of solutions for shape optimization problems of spectral type with both measure and perimeter constraints
Local minimality results for the mumford-shah functional via monotonicity
Full text linkLet Ω be a bounded piecewise regular open set with convex corners, and let MS(u) be the Mumford-Shah functional on the space SBV(Ω). We prove that the elastic minimizer of MS is a local minimizer with respect to the L1-topology. This is obtained as an application of interior and boundary monotonicity formulas for a weak notion of quasiminimizers of the Mumford-Shah energy. The local minimality result is then extended to more general free discontinuity problems taking into account also boundary conditions
L∞ bounds of Steklov eigenfunctions and spectrum stability under domain variation
No full textWe give a practical tool to control the L∞-norm of the Steklov eigenfunctions in a Lipschitz domain in terms of the norm of the BV-trace operator. The norm of this operator has the advantage to be characterized by purely geometric quantities. As a consequence, we give a spectral stability result for the Steklov eigenproblem under geometric domain perturbations and several examples where stability occurs. In particular we deal with geometric domains which are not equi-Lipschitz, like vanishing holes, merging sets, approximations of inner peaks
Degenerate Free Discontinuity Problems and Spectral Inequalities in Quantitative Form
No full textWe introduce a new geometric–analytic functional that we analyse in the context of free discontinuity problems. Its main feature is that the geometric term (the length of the jump set) appears with a negative sign. This is motivated by searching quantitative inequalities for the best constants of Sobolev–Poincaré inequalities with trace terms in Rn which correspond to fundamental eigenvalues associated to semilinear problems for the Laplace operator with Robin boundary conditions. Our method is based on the study of this new, degenerate, functional which involves an obstacle problem in interaction with the jump set. Ultimately, this becomes a mixed free discontinuity/free boundary problem occuring above/at the level of the obstacle, respectively
The multiphase mumford-shah problem
No full textWe perform a rigorous analysis of the multiphase version of the Mumford--Shah functional. A characteristic property of the formulation is the presence of a true partition of the image (so in two dimensions of closed contours), each cell of the partition possibly containing inner jumps. The nontrivial partitioning naturally occurs as a consequence of the presence in the energy functional of statistical terms or of phase dependent weights. In particular, we prove a multiphase version of the De Giorgi--Carriero--Leaci result
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