Related papers: Localization of eigenfunctions via an effective po…

We prove that the random Schrodinger operators on $\mathbb{R}^3$ with independent, identically distributed random variables and single-site potentials given by $\delta$-functions on $\mathbb{Z}^3$, exhibit both dynamical localization and…

We consider the Laplace operator with Dirichlet boundary conditions on a domain in R^d and study the effect that performing a scaling in one direction has on the eigenvalues and corresponding eigenfunctions as a function of the scaling…

Let $e_\l(x)$ be a Neumann eigenfunction with respect to the positive Laplacian $\Delta$ on a compact Riemannian manifold $M$ with boundary such that $\Delta\, e_\l=\l^2 e_\l$ in the interior of $M$ and the normal derivative of $e_\l$…

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…

In this paper, energy function is used to investigate the eigen-solutions of $-\Delta u+ Vu=\lambda u$ on the Riemannian manifolds. We give a new way to prove the positivity of the initial energy of energy function, which leads to a simple…

The hot spots conjecture of J. Rauch states that the second Neumann eigenfunction of the Laplace operator on a bounded Lipschitz domain in $\mathbb{R}^n$ attains its extrema only on the boundary of the domain. We present an analogous…

In this article we consider a special type of degenerate elliptic partial differential equations of second order in convex domains that satisfy the interior sphere condition. We show that any positive viscosity solution $u$ of $-|\nabla…

In this work, we revisit the following estimate due to Dahlberg \cite{Dahl}. Let $\textit{\textbf x}_0$ a fixed point in a bounded Lipschitz domain $\Omega$. Then there exists a constant $C > 0$ such that if $u$ is a harmonic function in…

We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann or Robin boundary condition. We keep the presentation at a level accessible to scientists from…

The paper deals with a comprehensive theory of mappings, whose local behavior can be described by means of linear subspaces, contained in the graphs of two (primal and dual) generalized derivatives. This class of mappings includes the…

This article is concerned with properties of delocalization for eigenfunctions of Schr\"odinger operators on large finite graphs. More specifically, we show that the eigenfunctions have a large support and we assess their lp-norms. Our…

In this article, we study the eigenvalue of nonlinear $p-$fractional Hardy operator \begin{align*} (-\Delta)_p^{\alpha}u - \mu \frac{|u|^{p-2}u}{|x|^{p\alpha}} = \lambda V(x) |u|^{p-2}u \; \text{in}\; \Omega, \quad u = 0 \; \mbox{in}\;…

Given a unital algebra $\mathscr A$ of locally Lipschitz functions defined over a metric measure space $({\mathrm X},{\mathsf d},\mathfrak m)$, we study two associated notions of function of bounded variation and their relations: the space…

Let $(M,g)$ be a compact Riemannian manifold with a boundary of class $\mathscr{C}^{1}$. We are interested in the spectrum of the weighted Laplacian on $M$ with Neumann boundary conditions. More precisely, given $\rho$ and $\sigma$ two…

On any metric space, I provide an intrinsic characterization of those complex-valued functions which are uniform limits of Lipschitz functions. There are applications to function theory on complete Riemannian manifolds and, in particular,…

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…

To minimize or upper-bound the value of a function "robustly", we might instead minimize or upper-bound the "epsilon-robust regularization", defined as the map from a point to the maximum value of the function within an epsilon-radius. This…

We consider the "Method of particular solutions" for numerically computing eigenvalues and eigenfunctions of the Laplacian $\Delta$ on a smooth, bounded domain Omega in RR^n with either Dirichlet or Neumann boundary conditions. This method…

We consider the Anderson model with Bernoulli potential on the 3D lattice, and prove localization of eigenfunctions corresponding to eigenvalues near zero, the lower boundary of the spectrum. We follow the framework by Bourgain-Kenig and…

An inverse problem of finding an obstacle and the boundary condition on its surface from the fixed-energy scattering data is studied. A new method is developed for a proof of the uniqueness results. The method does not use the discreteness…