Related papers: On Pleijel's nodal domain theorem for the Robin pr…

In this paper, we use a weighted isoperimetric inequality to give a lower bound on the first Dirichlet eigenvalue of the Laplacian on a bounded domain inside a Euclidean cone. Our bound is sharp, in that only sectors realize it. This result…

The inverse Robin problem covers the determination of the Robin parameter in an elliptic partial differential equation posed on a domain $\Omega$. Given the solution of the Robin problem on a subdomain $\omega \subset \Omega$ together with…

We consider a semilinear elliptic equation with Dirichlet boundary conditions in a smooth, possibly unbounded, domain. Under suitable assumptions, we deduce a condition on the size of the domain that implies the existence of a positive…

In this paper we address the uniqueness issue in the classical Robin inverse problem on a Lipschitz domain $\Omega\subset\RR^n$, with $L^\infty$ Robin coefficient, $L^2$ Neumann data and isotropic conductivity of class $W^{1,r}(\Omega)$,…

By developing the method of multipliers, we establish sufficient conditions which guarantee the total absence of eigenvalues of the Laplacian in the half-space, subject to variable complex Robin boundary conditions. As a further application…

In this paper, we prove two results for the Robin eigenvalue problem. One is an upper bound for the ratio of the first two eigenvalues which can be used to recover the PPW conjecture proved by M.S.Ashbaugh and R.D.Benguria, the other is a…

We show that as the ratio between the first Dirichlet eigenvalues of a convex domain and of the ball with the same volume becomes large, the same must happen to the corresponding ratio of isoperimetric constants. The proof is based on the…

We investigate regularity properties of the $\overline{\partial}$-equation on domains in a complex euclidean space that depend on a parameter. Both the interior regularity and the regularity in the parameter are obtained for a continuous…

We prove that every metric graph which is a tree has an orthonormal sequence of Laplace-eigenfunctions of full support. This implies that the number of nodal domains $\nu_n$ of the $n$-th eigenfunction of the Laplacian with standard…

In \cite{CJ1} M. Jaoua et al. studied the linear approximation of Robin problem on $\Omega$ an open bounded domain of $\R^d$, and they given some important results. In this paper, we study a nonlinear approximation of an elliptic problem…

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…

This paper is devoted to the refine analysis of Courant's theorem for the Dirichlet Laplacian. Many papers (and some of them quite recent) have investigated in which cases this inequality in Courant's theorem is an equality: Pleijel,…

Let $(M, g)$ be a closed Riemannian manifold, where g is $C^1$-smooth metric. Consider the sequence of eigenfunctions $u_k$ of the Laplace operator on M. Let $B$ be a ball on $M$. We prove a sharp estimate of the number of nodal domains of…

The first two eigenvalues of the Robin Laplacian are investigated along with their gap and ratio. Conjectures by various authors for arbitrary domains are supported here by new results for rectangular boxes. Results for rectangular domains…

We prove P\'olya's conjecture for the eigenvalues of the Dirichlet Laplacian on annular domains. Our approach builds upon and extends the methods we previously developed for disks and balls. It combines variational bounds, estimates of…

This article tackles the spectral analysis of the Robin Laplacian on a smooth bounded two-dimensional domain in the presence of a constant magnetic field. In the semiclassical limit, a uniform description of the spectrum located between the…

In this paper we study a Steklov-Robin eigenvalue problem for the Laplacian in annular domains. More precisely, we consider $\Omega=\Omega_0 \setminus \overline{B}_{r}$, where $B_{r}$ is the ball centered at the origin with radius $r>0$ and…

The hybrid spectral problem where the field satisfies Dirichlet conditions (D) on part of the boundary of the relevant domain and Neumann (N) on the remainder is discussed in simple terms. A conjecture for the C_1 coefficient is presented…

The probabilistic representation of weak solutions to a parabolic boundary value problem is established in the following framework. The boundary value problem consists of a second order parabolic equation defined on a time-varying Lipschitz…

In this paper we deal with spectral optimization for the Robin Laplacian on a family of planar domains admitting parallel coordinates, namely a fixed-width strip built over a smooth closed curve and the exterior of a convex set with a…