Related papers: A lower bound for the nodal sets of Steklov eigenf…

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…

We present a unified description of extremal metrics for the Laplace and Steklov eigenvalues on manifolds of arbitrary dimension using the notion of $n$-harmonic maps. Our approach extends the well-known results linking extremal metrics for…

In this paper, we successfully establish a Courant-type nodal domain theorem for both the Dirichlet eigenvalue problem and the closed eigenvalue problem of the Witten-Laplacian. Moreover, we also characterize the properties of the nodal…

This is a survey of recent results on eigenfunctions of the Laplacian on compact Riemannian manifolds and their nodal sets. It is the write-up of my talk at JDG 2011.

We study the high-frequency behavior of the Dirichlet-to-Neumann map for an arbitrary compact Riemannian manifold with a non-empty smooth boundary. We show that far from the real axis it can be approximated by a simpler operator. We use…

We prove that, given any knot $\gamma$ in a compact 3-manifold M, there exists a Riemannian metric on M such that there is a complex-valued eigenfunction u of the Laplacian, corresponding to the first nontrivial eigenvalue, whose nodal set…

We give lower bounds for the fundamental tone of open sets in submanifolds with locally bounded mean curvature in $ N \times \mathbb{R}$, where $N$ is an $n$-dimensional complete Riemannian manifold with radial sectional curvature $K_{N}…

We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact $n$-dimensional manifold which has nonnegative Ricci curvature and strictly convex boundary. When $n=3$, this implies…

In this article we consider the following weighted nonlinear eigenvalue problem for the $g-$Laplacian $$ -\mathop{\text{ div}}\left( g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right) = \lambda w(x) h(|u|)\frac{u}{|u|} \quad \text{ in…

In this paper, we study the higher Steklov eigenvalues of graphs on surfaces. We obtain the upper bound of higher Steklov eigenvalues of a finite graph $G$ with boundary $B$ and genus $g$ by using metrical deformation via probability flows.…

We introduce a graph $\Gamma$ which is roughly isometric to the hyperbolic plane and we study the Steklov eigenvalues of a subgraph with boundary $\Omega$ of $\Gamma$. For $(\Omega_l)_{l\geq 1}$ a sequence of subraphs of $\Gamma$ such that…

We consider the first positive Steklov eigenvalue on planar domains. First, we provide an example of a planar domain for which a first eigenfunction has a closed nodal line. Second, we establish a lower bound for the first positive…

We numerically compute the lowest Laplacian eigenvalues of several two-dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our approach is based on the method of particular solutions with domain decomposition. We…

We study the asymptotic behavior of Dirichlet minimizers to the Allen--Cahn equation on manifolds with boundary, and we relate the Neumann data to the geometry of the boundary. We show that Dirichlet minimizers are asymptotically local in…

Let $(M,g)$ be a compact, smooth Riemannian manifold and $\{u_h\}$ be a sequence of $L^2$-normalized Laplace eigenfunctions that has a localized defect measure $\mu$ in the sense that $ M \setminus \text{supp}(\pi_* \mu) \neq \emptyset$…

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold without boundary and $e_\lambda$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator with respect to the metric $g$, i.e \[ -\Delta_g e_\lambda = \lambda^2…

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 study the existence of Neumann eigenfunctions which do not change sign on the boundary of some special domains. We show that eigenfunctions which are strictly positive on the boundary exist on regular polygons with at least 5 sides,…

We study an eigenvalue problem for the biharmonic operator with Neumann boundary conditions on domains of Riemannian manifolds. We discuss the weak formulation and the classical boundary conditions, and we describe a few properties of the…

A Laplacian eigenfunction on a two-dimensional manifold dictates some natural partitions of the manifold; the most apparent one being the well studied nodal domain partition. An alternative partition is revealed by considering a set of…