Related papers: Local and global analysis of eigenfunctions

These notes present a recent approach to study the high-frequency eigenstates of the Laplacian on compact Riemannian manifolds of negative sectional curvature. The main result is a lower bound on the Kolmogorov-Sinai entropy of the…

We study Neumann functions for divergence form, second order elliptic systems with bounded measurable coefficients in a bounded Lipschitz domain or a Lipschitz graph domain. We establish existence, uniqueness, and various estimates for the…

We give estimates for the $L^p$ norm ($2\leq p \leq +\infty$) of the restriction to a curve of the eigenfunctions of the Laplace Beltrami operator on a Riemannian surface. If the curve is a geodesic, we show that on the sphere these…

In this paper, we prove some isoperimetric bounds for lower order eigenvalues of the Wentzell-Laplace operator on bounded domains of a Euclidean space or a Hadamard manifold, of the Laplacian on closed hypersurfaces of a Euclidean space or…

This paper explores the existence and properties of \emph{basic} eigenvalues and eigenfunctions associated with the Riemannian Laplacian on closed, connected Riemannian manifolds featuring an effective isometric action by a compact Lie…

We present a general framework, treating Lipschitz domains in Riemannian manifolds, that provides conditions guaranteeing the existence of norming sets and generalized local polynomial reproduction - a powerful tool used in the analysis of…

We establish lower bounds for the first non-zero eigenvalue for the natural geometric sub-elliptic Laplacian operator defined on sub-Riemannian manifolds of step 2 that satisfy a positive curvature condition. The methods are very general…

We prove sharp $L^p$ estimates for the Steklov eigenfunctions on compact manifolds with boundary in terms of their $L^2$ norms on the boundary. We prove it by establishing $L^p$ bounds for the harmonic extension operators as well as the…

We study Laplacians associated to a graph and single out a class of such operators with special regularity properties. In the case of locally finite graphs, this class consists of all selfadjoint, non-negative restrictions of the standard…

In this paper we study the eigenvalue problems for a nonlocal operator of order $s$ that is analogous to the local pseudo $p-$Laplacian. We show that there is a sequence of eigenvalues $\lambda_n \to \infty$ and that the first one is…

In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary…

In the present paper several bounds on multiplicities of eigenvalues of the Laplacian operator on surfaces are generalized from the case of either closed surface or simply-connected planar domain to the case of a surface of positive genus…

Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{\phi_\lambda \}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-\Delta_g\phi_\lambda =\lambda^2 \phi_\lambda$. Given a smooth submanifold $H \subset M$ of codimension…

Let $e_\l(x)$ be an eigenfunction with respect to the Dirichlet Laplacian $\Delta_N$ on a compact Riemannian manifold $N$ with boundary: $\Delta_N e_\l=\l^2 e_\l$ in the interior of $N$ and $e_\l=0$ on the boundary of $N$. We show the…

This paper concerns the behavior of the eigenfunctions and eigenvalues of the round sphere's Laplacian acting on the space of sections of a real line bundle which is defined on the complement of an even numbers of points in $S^2$. Of…

Let $(M,g)$ be a compact Riemannian surface. Consider a family of $L^2$ normalized Laplace-Beltrami eigenfunctions, written in the semiclassical form $-h_j^2\Delta_g \phi_{h_j} = \phi_{h_j}$, whose eigenvalues satisfy $h h_j^{-1} \in (1, 1…

We study the existence of principal eigenvalues and principal eigenfunctions for weighted eigenvalue problems of the form: \begin{equation*} - \mbox{div} ( L (x) |\nabla u|^{p-2} \nabla u ) = \lambda K(x) |u|^{p-2} u \hspace{.1cm} \mbox {…

We consider the problem of proving $L^p$ bounds for eigenfunctions of the Laplacian in the high frequency limit in the presence of nonpositive curvature and more generally, manifolds without conjugate points. In particular, we prove…

Let $(M,g)$ be a complete non-compact Riemannian manifold together with a function $e^h$, which weights the Hausdorff measures associated to the Riemannian metric. In this work we assume lower or upper radial bounds on some weighted or…

In this paper we consider a non-local problem for a Laplace operator in a multidimensional bounded symmetric domain. The investigated problem is an analogue of the classical periodic boundary value problems in the case of non-rectangular…