Related papers: The exterior Steklov problem for Euclidean domains
We prove Reilly-type upper bounds for the first non-zero eigenvalue of the Steklov problem associated with the $p$-Laplace operator on submanifolds of manifolds with sectional curvature bounded form above by a non-negative constant.
We consider the Stokes eigenvalue problem in open balls and open annuli in R3 with homogeneous Dirichlet boundary conditions. Using the frame of toroidal and poloidal fields we construct the othogonal decomposition of the Stokes eigenvalue…
We develop a numerical method for solving shape optimization of functionals involving Steklov eigenvalues and apply it to the problem of maximization of the $k$-th Steklov eigenvalue, under volume constraint. A similar study in the planar…
We prove two explicit bounds for the multiplicities of Steklov eigenvalues $\sigma_k$ on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index $k$ of an eigenvalue, while the other depends as…
We generalize an inequality for the determinant of a real matrix proved by A. Schinzel, to more general exterior products of vectors in Euclidean space. We apply this inequality to the logarithmic embedding of $S$-units contained in a…
The paper is concerned with the doubling estimates and vanishing order of the Steklov eigenfunction on the boundary of a smooth boundary domain $\mathbb R^n$. The eigenfunction is given by a Dirichlet-to-Neumann map. We improve the doubling…
We consider three different questions related to the Steklov and mixed Steklov problems on surfaces. These questions are connected by the techniques that we use to study them, which exploit symmetry in various ways even though the surfaces…
We study the geometry of the first two eigenvalues of a magnetic Steklov problem on an annulus $\Sigma$ (a compact Riemannian surface with genus zero and two boundary components), the magnetic potential being the harmonic one-form having…
We prove new existence and uniqueness results in full Sobolev spaces for the steady-state Oseen problem in a smooth exterior domain of $\mathbb{R}^n$, $n\ge 2$. These results are then employed, on the one hand, in the study of analogous…
In this note, we investigate the Steklov spectrum of the warped product $[0,L]\times_h \Sigma$ equipped with the metric $dt^2+h(t)^2g_\Sigma$, where $\Sigma$ is a compact surface. We find sharp upper bounds for the Steklov eigenvalues in…
We address the Mach limit problem for the Euler equations in an exterior domain with analytic boundary. We first prove the existence of tangential analytic vector fields for the exterior domain with constant analyticity radii, and introduce…
On any compact manifold of dimension $n\geq3$ with boundary, we prescibe any finite part of the Steklov spectrum whithin a given conformal class. In particular, we prescribe the multiplicity of the first eigenvalues. On a compact surface…
We establish rigorous quantitative inequalities for the first eigenvalue of the generalized $p$-Robin problem, for both the classical diffusion absorption case, where the Robin boundary parameter $\alpha$ is positive, and the…
Motivated by the connection between the first eigenvalue of the Dirichlet-Laplacian and the torsional rigidity, the aim of this paper is to find a physically coherent and mathematically interesting new concept for boundary torsional…
In Euclidean and Hyperbolic space, and the hemisphere in $S^n$, geodesic balls maximize the gap $\lambda_2 - \lambda_1$ of Dirichlet eigenvalues, amoung domains with fixed $\lambda_1$. We prove an upper bound on $\lambda_2 - \lambda_1$ for…
The Faber-Krahn inequality states that the first Dirichlet eigenvalue among all bounded domains is no less than a Euclidean ball with the same volume in $\mathbb{R}^n$ \cite{Chavel FB}. B{\i}y{\i}ko\u{g}lu and Leydold (J. Comb. Theory, Ser.…
We justify the Weyl asymptotic formula for the eigenvalues of the Poincar\'e-Steklov spectral problem for a domain bounded by a Lipschitz surface.
We consider the nonlinear Neumann eigenvalue problem in outward cuspidal domains with a weighted measure. Using composition operators on Sobolev spaces, we establish embeddings of Sobolev spaces into weighted Lebesgue spaces. These…
In this paper, we consider the Hessian equations in some exterior domain with prescribed asymptotic behavior at infinity and Dirichlet-Neumann conditions on its interior boundary. We obtain that there exists a unique bounded domain such…
In this paper, we mainly study eigenvalue problems of p-Laplacian on domains with an interior hole. Firstly we prove Faber-Krahn-type inequalities, and Cheng-type eigenvalue comparison theorems on manifolds. Secondly, we prove a comparison…