Related papers: Second variation of domain functionals and applica…

Given the eigenvalue problem for the Laplacian with Robin boundary conditions, (with $\beta\in\R\setminus\{0\}$ the Robin parameter), we consider a shape minimization problem for a function of the first eigenvalues if $\beta>0$ and a shape…

We show that the ball does not maximize the first nonzero Steklov eigenvalue among all contractible domains of fixed boundary volume in $\mathbb{R}^n$ when $n \geq 3$. This is in contrast to the situation when $n=2$, where a result of…

We study the biharmonic Steklov eigenvalue problem on a compact Riemannian manifold $\Omega$ with smooth boundary. We give a computable, sharp lower bound of the first eigenvalue of this problem, which depends only on the dimension, a lower…

We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifold with boundary. These domains are close to half balls of small radius centered at a…

We study the discretization of an elliptic partial differential equation, posed on a two- or three-dimensional domain with smooth boundary, endowed with a generalized Robin boundary condition which involves the Laplace-Beltrami operator on…

We prove that among all doubly connected domains of $\mathbb{R}^n$ of the form $B_1\backslash \overline{B_2}$, where $B_1$ and $B_2$ are open balls of fixed radii such that $\overline{B_2}\subset B_1$, the first nonzero Steklov eigenvalue…

In this paper, we deal with functionals involving the torsion and the first eigenvalue of the Laplacian with Robin boundary conditions (to which we refer as Robin Torsion and Robin Eigenvalue), with other geometrical quantities, in the…

We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Wentzell-Laplace operator of a domain $\Omega$, involving only geometrical informations. We provide such an upper bound, by generalizing Brock's…

Using three different notions of generalized principal eigenvalue of linear second order elliptic operators in unbounded domains, we derive necessary and sufficient conditions for the validity of the maximum principle, as well as for the…

We investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness and ellipticity on the coefficient function and very mild conditions on the geometry of the…

We consider Steklov eigenvalues of nearly hyperspherical domains in $\mathbb{R}^{d + 1}$ with $d\ge 3$. In previous work, treating such domains as perturbations of the ball, we proved that the Steklov eigenvalues are analytic functions of…

In the present paper, we study the variational properties of Steklov transmission eigenvalues, which can be seen as eigenvalues of the sum of two Dirichlet-to-Neumann operators on two different sides of a given curve contained in a surface.…

We study the Robin boundary-value problem for bounded domains with isolated singularities. Because for such domains trace spaces of space $H^1(D)$ on its boundaries are weighted Sobolev spaces $L^{2, \xi}(\partial D)$ existence and…

We consider second order elliptic operators with real, nonsymmetric coefficient functions which are subject to mixed boundary conditions. The aim of this paper is to provide uniform resolvent estimates for the realizations of these…

The Steklov problem on a compact Lipschitz domain is to find harmonic functions on the interior whose outward normal derivative on the boundary is some multiple (eigenvalue) of its trace on the boundary. These eigenvalues form the Steklov…

We consider Steklov eigenvalues on nearly spherical and nearly annular domains in $d$ dimensions. By using the Green-Beltrami identity for spherical harmonic functions, the derivatives of Steklov eigenvalues with respect to the domain…

We consider a classical shape optimization problem for the eigenvalues of elliptic operators with homogeneous boundary conditions on domains in the $N$-dimensional Euclidean space. We survey recent results concerning the analytic dependence…

We study three types of fourth-order Steklov eigenvalue problems. For the first two of them, we derive the asymptotic expansion of their spectra on Euclidean annular domains $\mathbb{B}^n_1\setminus \overline{\mathbb{B}^n_\epsilon}$ as…

We study the solvability of boundary-value problems for differential-operator equations of the second order in L p (0, 1; X), with 1 < p < +$\infty$, X being a UMD complex Banach space. The originality of this work lies in the fact that we…

The goal of this paper is to derive in the two-dimensional case necessary and sufficient minimality conditions in terms of the second variation for the functional \[ v\mapsto\int_{\Omega}\big(|\nabla v|^{2}+\chi_{\{v>0\}}Q^{2} \big)\,dx, \]…