Search results for "Laplace operator"
showing 10 items of 101 documents
Correlation at Low Temperature: II. Asymptotics
2004
The present paper is a continuation of ref. 4, where the truncated two-point correlation function for a class of lattice spin systems was proved to have exponential decay at low temperature, under a weak coupling assumption. In this paper we compute the asymptotics of the correlation function as the temperature goes to zero. This paper thus extends ref. 3 in two directions: The Hamiltonian function is allowed to have several local minima other than a unique global minimum, and we do not require translation invariance of the Hamiltonian function. We are in particular able to handle spin systems on a general lattice.
Continuous spectrum for a two phase eigenvalue problem with an indefinite and unbounded potential
2020
Abstract We consider a two phase eigenvalue problem driven by the ( p , q ) -Laplacian plus an indefinite and unbounded potential, and Robin boundary condition. Using a modification of the Nehari manifold method, we show that there exists a nontrivial open interval I ⊆ R such that every λ ∈ I is an eigenvalue with positive eigenfunctions. When we impose additional regularity conditions on the potential function and the boundary coefficient, we show that we have smooth eigenfunctions.
An overdetermined problem for the anisotropic capacity
2015
We consider an overdetermined problem for the Finsler Laplacian in the exterior of a convex domain in \({\mathbb {R}}^{N}\), establishing a symmetry result for the anisotropic capacitary potential. Our result extends the one of Reichel (Arch Ration Mech Anal 137(4):381–394, 1997), where the usual Newtonian capacity is considered, giving rise to an overdetermined problem for the standard Laplace equation. Here, we replace the usual Euclidean norm of the gradient with an arbitrary norm H. The resulting symmetry of the solution is that of the so-called Wulff shape (a ball in the dual norm \(H_0\)).
Atmospheric Turbulence Effects Removal on Infrared Sequences Degraded by Local Isoplanatism
2007
When observing an object horizontally at a long distance, degradations due to atmospheric turbulence often occur. Different methods have already been tested to get rid of this kind of degradation, especially on infrared sequences. It has been shown that the Wiener filter applied locally on each frame of a sequence allows to obtain good results in terms of edges, while the regularization by the Laplacian operator applied in the same way provides good results in terms of noise removal in uniform areas. In this article, we present hybrid methods which take advantages of both Wiener filter and Laplacian regularization.
Higher integrability and stability of (p,q)-quasiminimizers
2023
Using purely variational methods, we prove local and global higher integrability results for upper gradients of quasiminimizers of a $(p,q)$-Dirichlet integral with fixed boundary data, assuming it belongs to a slightly better Newtonian space. We also obtain a stability property with respect to the varying exponents $p$ and $q$. The setting is a doubling metric measure space supporting a Poincar\'e inequality.
Reilly's type inequality for the Laplacian associated to a density related with shrinkers for MCF
2015
Let $(\bar{M},,e^\psi)$ be a Riemannian manifold with a density, and let $M$ be a closed $n$-dimensional submanifold of $\bar{M}$ with the induced metric and density. We give an upper bound on the first eigenvalue $\lambda_1$ of the closed eigenvalue problem for $\Delta_\psi$ (the Laplacian on $M$ associated to the density) in terms of the average of the norm of the vector ${\vec{H}}_{{\psi}} + {\bar \nabla}$ with respect to the volume form induced by the density, where ${\vec{H}}_{{\psi}}$ is the mean curvature of $M$ associated to the density $e^\psi$. When $\bar{M}=\Bbb R^{n+k}$ or $\bar{M}=S^{n+k-1}$, the equality between $\lambda_1$ and its bound implies that $e^\psi$ is a Gaussian den…
Partial data inverse problems for the Hodge Laplacian
2017
We prove uniqueness results for a Calderon type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth order potential. The method is based on Carleman estimates for the Hodge Laplacian with relative or absolute boundary conditions, and on the construction of complex geometric optics solutions which reduce the Calderon type problem to a tensor tomography problem for 2-tensors. The arguments in this paper allow to establish partial data results for elliptic systems that generalize the scalar resu…
Fokker–Planck equation with respect to heat measures on loop groups
2011
Abstract The Dirichlet form on the loop group L e ( G ) with respect to the heat measure defines a Laplacian Δ DM on L e ( G ) . In this note, we will use Wasserstein distance variational method to solve the associated heat equation for a given data of finite entropy.
Homoclinic Solutions of Nonlinear Laplacian Difference Equations Without Ambrosetti-Rabinowitz Condition
2021
The aim of this paper is to establish the existence of at least two non-zero homoclinic solutions for a nonlinear Laplacian difference equation without using Ambrosetti-Rabinowitz type-conditions. The main tools are mountain pass theorem and Palais-Smale compactness condition involving suitable functionals.
Removable sets for continuous solutions of quasilinear elliptic equations
2001
We show that sets of n − p + α ( p − 1 ) n-p+\alpha (p-1) Hausdorff measure zero are removable for α \alpha -Hölder continuous solutions to quasilinear elliptic equations similar to the p p -Laplacian. The result is optimal. We also treat larger sets in terms of a growth condition. In particular, our results apply to quasiregular mappings.