Pier Domenico Lamberti

Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators

We prove sharp stability estimates for the variation of the eigenvalues of non-negative self-adjoint elliptic operators of arbitrary even order upon variation of the open sets on which they are defined. These estimates are expressed in terms of the Lebesgue measure of the symmetric difference of the open sets. Both Dirichlet and Neumann boundary conditions are considered.

Spectral Stability of Higher Order Uniformly Elliptic Operators

We prove estimates for the variation of the eigenvalues of uniformly elliptic operators with homogeneous Dirichlet or Neumann boundary conditions upon variation of the open set on which an operator is defined. We consider operators of arbitrary even order and open sets admitting arbitrary strong degeneration. The main estimate is expressed in terms of a natural and easily computable distance between open sets with continuous boundaries. Another estimate is obtained in terms of the lower Hausdorff—Pompeiu deviation of the boundaries, which in general may be much smaller than the usual Hausdorff—Pompeiu distance. Finally, in the case of diffeomorphic open sets, we obtain an estimate even without the assumption of continuity of the boundaries.

Viewing the Steklov eigenvalues of the Laplace operator as critical Neumann eigenvalues

We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of boundary mass concentration. We discuss the asymptotic behavior of the Neumann eigenvalues in a ball and we deduce that the Steklov eigenvalues minimize the Neumann eigenvalues. Moreover, we study the dependence of the eigenvalues of the Steklov problem upon perturbation of the mass density and show that the Steklov eigenvalues violates a maximum principle in spectral optimization problems.

A Global Lipschitz Continuity Result for a Domain Dependent Dirichlet Eigenvalue Problem for the Laplace Operator

Let Ω be an open connected subset of Rn for which the Poincare inequality holds. We consider the Dirichlet eigenvalue problem for the Laplace operator in the open subset φ(Ω) of Rn, where φ is a locally Lipschitz continuous homeomorphism of Ω onto φ(Ω). Then we show Lipschitz type inequalities for the reciprocals of the eigenvalues of the Rayleigh quotient∫ φ(Ω) |Dv| 2 dy ∫ φ(Ω) |v|2 dy upon variation of φ, which in particular yield inequalities for the proper eigenvalues of the Dirichlet Laplacian when we further assume that the imbedding of the Sobolev space W 1,2 0 (Ω) into the space L 2(Ω) is compact. In this case, we prove the same type of inequalities for the projections onto the eigenspaces upon variation of φ.

AN ANALYTICITY RESULT FOR THE DEPENDENCE OF MULTIPLE EIGENVALUES AND EIGENSPACES OF THE LAPLACE OPERATOR UPON PERTURBATION OF THE DOMAIN

In this paper, we consider the dependence of the Dirichlet eigenva- lues and eigenspaces of the Laplace operator upon perturbation of the domain of definition. We prove that the dependence of a certain eigenvalue and of the corre- sponding eigenspace is analytic on the set of perturbations that leave the multiplicity constant.

Stability Estimates for Resolvents, Eigenvalues, and Eigenfunctions of Elliptic Operators on Variable Domains

We consider general second order uniformly elliptic operators sub- ject to homogeneous boundary conditions on open sets `(›) parametrized by Lipschitz homeomorphismsdeflned on a flxed reference domain ›. For two open sets `(›) and e `(›) we estimate the variation of resolvents, eigenvalues, and eigenfunctions via the Sobolev norm ke ` i `kW 1;p(›) for flnite values of p, under natural summability conditions on eigenfunctions and their gradi- ents. We prove that such conditions are satisfled for a wide class of operators and open sets, including open sets with Lipschitz continuous boundaries. We apply these estimates to control the variation of the eigenvalues and eigen- functions via the measure of the symmetric difierence of the open sets. We also discuss an application to the stability of solutions to the Poisson problem.

Steklov-type eigenvalues associated with best Sobolev trace constants: domain perturbation and overdetermined systems

We consider a variant of the classic Steklov eigenvalue problem, which arises in the study of the best trace constant for functions in Sobolev space. We prove that the elementary symmetric functions of the eigenvalues depend real-analytically upon variation of the underlying domain and we compute the corresponding Hadamard-type formulas for the shape derivatives. We also consider isovolumetric and isoperimetric domain perturbations and we characterize the corresponding critical domains in terms of appropriate overdetermined systems. Finally, we prove that balls are critical domains for the elementary symmetric functions of the eigenvalues subject to volume or perimeter constraint.

Spectral stability estimates for elliptic operators subject to domain transformations with non-uniformly bounded gradients

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain

Higher order elliptic operators on variable domains. Stability results and boundary oscillations for intermediate problems

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Neumann to Steklov eigenvalues: asymptotic and monotonicity results

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