Related papers: Sharp bounds for higher Steklov-Dirichlet eigenval…

Consider a compact Riemannian manifold with boundary. In this short note we prove that under certain positive curvature assumptions on the manifold and its boundary the Steklov eigenvalues of the manifold are controlled by the Laplace…

We consider Steklov eigenvalues of nearly circular domains in $\R^{2}$ of fixed unitary area. In \cite{viator2018}, the authors treated such domains as perturbations of the disk, and they computed the first-order term of the asymptotic…

Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace eigenvalues of that manifold. The domains are obtained…

Let M be a compact Riemannian manifold with boundary. Let b>0 be the number of connected components of its boundary. For manifolds of dimension at least 3, we prove that it is possible to obtain an arbitrarily large (b+1)-th Steklov…

First we establish a weighted Reilly formula for differential forms on a smooth compact oriented Riemannian manifold with boundary. Then we give two applications of this formula when the manifold satisfies certain geometric conditions. One…

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…

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 consider the Steklov-Dirichlet eigenvalue problem on eccentric annuli in Euclidean space of general dimensions. In recent work by the same authors of this paper [21], a limiting behavior of the first eigenvalue, as the distance between…

We consider the Steklov problem on differential $p$-forms defined by M. Karpukhin and present geometric eigenvalue bounds in the setting of warped product manifolds in various scenarios. In particular, we obtain Escobar type lower bounds…

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 study a class of Riemannian manifolds which are equipped with a singular metric. In particular we study a domain perturbation problem for the Dirichlet eigenvalues which depends on the best constant in the Hardy Inequality. However, we…

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…

In this note we present upper bounds for the variational eigenvalues of the Steklov $p$-Laplacian on domains of $\mathbb R^n$, $n\geq 2$. We show that for $1<p\leq n$ the variational eigenvalues $\sigma_{p,k}$ are bounded above in terms of…

We show that for compact Riemannian manifolds of dimension at least $3$ with nonempty boundary, we can modify the manifold by performing surgeries of codimension $2$ or higher, while keeping the Steklov spectrum nearly unchanged. This shows…

In this note we establish an expression for the Steklov spectrum of warped products in terms of auxiliary Steklov problems for drift Laplacians with weight induced by the warping factor. As an application, we show that a compact manifold…

We give some sharp lower bounds of the first eigenvalue for the Hodge Laplacian acting on differential forms on the boundary of a Riemannian manifold. We also give some sharp estimates for the first nonzero Steklov eigenvalue for…

We establish a new lower bound for the first non-zero Steklov eigenvalue of a compact Riemannian manifold with non-negative Ricci curvature and (strictly) convex boundary. Related results are also obtained under weaker geometric hypotheses.

We consider a shape optimization problem for the first mixed Steklov-Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. We give a geometric proof which is motivated by Newton's shell theorem

In this paper, sharp bounds for the first nonzero eigenvalues of different type have been obtained. Moreover, when those bounds are achieved, related rigidities can be characterized. More precisely, first, by applying the Bishop-type volume…

We study the following class of Steklov eigenvalue problems: \[ \nabla \cdot \bigl( w \nabla u \bigr) = 0 \quad \text{in } \Omega, \qquad \frac{\partial u}{\partial \nu} = \gamma v u \quad \text{on } \partial \Omega, \] where $w$ and $v$…