Related papers: Weyl asymptotics for Poincar\'{e}-Steklov eigenval…

We prove a two-term Weyl-type asymptotic formula for sums of eigenvalues of the Dirichlet Laplacian in a bounded open set with Lipschitz boundary. Moreover, in the case of a convex domain we obtain a universal bound which correctly…

In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of biharmonic Stekloff eigenvalues with Neumann boundary condition in a bounded domain of an $n$-dimensional Riemannian manifold.

We prove an asymptotic estimate for the growth of variational eigenvalues of fractional p-Laplacian eigenvalue problems on a smooth bounded domain.

The validity of Weyl's law for the Steklov problem on domains with Lipschitz boundaries is a well-known open question in spectral geometry. We answer this question in two dimensions and show that Weyl's law holds for an even larger class of…

We give a simple proof of the Weyl asymptotic formula for eigenvalues of the Dirichlet Laplacian, the buckling problem, and the Dirichlet bi-Laplacian in Euclidean domains of finite volume, with no assumptions about the boundary.

We obtain precise asymptotics for the Steklov eigenvalues on a compact Riemannian surface with boundary. It is shown that the number of connected components of the boundary, as well as their lengths, are invariants of the Steklov spectrum.…

We provide a direct proof of Weyl's law for the buckling eigenvalues of the biharmonic operator on a wide class of domains of $\mathbb R^d$ including bounded Lipschitz domains. The proof relies on asymptotically sharp lower and upper bounds…

We consider a general second-order elliptic differential operator on a domain with a cylindrical end. We impose Dirichlet boundary conditions on the boundary with the exception of a small set, where we impose Neumann boundary conditions.…

We present upper and lower bounds for Steklov eigenvalues for domains in $\mathbb{R}^{N+1}$ with $C^2$ boundary compatible with the Weyl asymptotics. In particular, we obtain sharp upper bounds on Riesz-means and the trace of corresponding…

We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading…

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…

The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension d greater or equal to 3. Apart…

Let $\Omega$ be a bounded domain with $C^2$-smooth boundary in an $n$-dimensional oriented Riemannian manifold. It is well-known that for the bi-harmonic equation $\Delta^2 u=0$ in $\Omega$ with the $0$-Dirichlet boundary condition, there…

In this paper, we investigate eigenvalues of the Wentzel-Laplace operator on a bounded domain in some Riemannian manifold. We prove asymptotically optimal estimates, according to the Weyl's law through bounds that are given in terms of the…

We consider Steklov eigenvalues of three-dimensional, nearly-spherical domains. In previous work, we have shown that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute the first-order term…

In this note we obtain an asymptotic estimate for growth behavior of variational eigenvalues of the $p-$fractional eigenvalue problem on a smooth bounded domain with Dirichlet boundary condition.

This paper is concerned with the discrete spectrum of the self-adjoint realization of the semi-classical Schr\"odinger operator with constant magnetic field and associated with the de Gennes (Fourier/Robin) boundary condition. We derive an…

We prove Weyl type of asymptotic formulas for the real and the complex internal transmission eigenvalues when the domain is a ball and the index of refraction is constant.

The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been…

We prove that in Riemannian manifolds the $k$-th Steklov eigenvalue on a domain and the square root of the $k$-th Laplacian eigenvalue on its boundary can be mutually controlled in terms of the maximum principal curvature of the boundary…