Search results for " laplacia"

The ∞-Eigenvalue Problem

1999

. The Euler‐Lagrange equation of the nonlinear Rayleigh quotient \( \left(\int_{\Omega}|\nabla u|^{p}\,dx\right) \bigg/ \left(\int_{\Omega}|u|^{p}\,dx\right)\) is \( -\div\left( |\nabla u|^{p-2}\nabla u \right)= \Lambda_{p}^{p} |u |^{p-2}u,\) where \(\Lambda_{p}^{p}\) is the minimum value of the quotient. The limit as \(p\to\infty\) of these equations is found to be \(\max \left\{ \Lambda_{\infty}-\frac{|\nabla u(x)|}{u(x)},\ \ \Delta_{\infty}u(x)\right\}=0,\) where the constant \(\Lambda_{\infty}=\lim_{p\to\infty}\Lambda_{p}\) is the reciprocal of the maximum of the distance to the boundary of the domain Ω.

Mathematical and numerical analysis of initial boundary valueproblem for a linear nonlocal equation

2019

We propose and study a numerical scheme for bounded distributional solutions of the initial boundary value problem for the anomalous diffusion equation ∂t u +Lμu = 0 in a bounded domain supplemented with inhomogeneous boundary conditions. Here Lμ is a class of nonlocal operators including fractional Laplacian. ⃝c 2019 InternationalAssociation forMathematics andComputers in Simulation (IMACS). Published by ElsevierB.V.All rights reserved.

C0-semigroups norm continuous at infinity

1996

Harnack's inequality for p-harmonic functions via stochastic games

2013

We give a proof of asymptotic Lipschitz continuity of p-harmonious functions, that are tug-of-war game analogies of ordinary p-harmonic functions. This result is used to obtain a new proof of Lipsc...

Principal eigenvalue of a very badly degenerate operator and applications

2007

Abstract In this paper, we define and investigate the properties of the principal eigenvalue of the singular infinity Laplace operator Δ ∞ u = ( D 2 u D u | D u | ) ⋅ D u | D u | . This operator arises from the optimal Lipschitz extension problem and it plays the same fundamental role in the calculus of variations of L ∞ functionals as the usual Laplacian does in the calculus of variations of L 2 functionals. Our approach to the eigenvalue problem is based on the maximum principle and follows the outline of the celebrated work of Berestycki, Nirenberg and Varadhan [H. Berestycki, L. Nirenberg, S.R.S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operator…

Unique continuation property and Poincar�� inequality for higher order fractional Laplacians with applications in inverse problems

2020

We prove a unique continuation property for the fractional Laplacian $(-\Delta)^s$ when $s \in (-n/2,\infty)\setminus \mathbb{Z}$. In addition, we study Poincar\'e-type inequalities for the operator $(-\Delta)^s$ when $s\geq 0$. We apply the results to show that one can uniquely recover, up to a gauge, electric and magnetic potentials from the Dirichlet-to-Neumann map associated to the higher order fractional magnetic Schr\"odinger equation. We also study the higher order fractional Schr\"odinger equation with singular electric potential. In both cases, we obtain a Runge approximation property for the equation. Furthermore, we prove a uniqueness result for a partial data problem of the $d$-…

A DUALITY APPROACH TO THE FRACTIONAL LAPLACIAN WITH MEASURE DATA

2011

We describe a duality method to prove both existence and uniqueness of solutions to nonlocal problems like $$(-\Delta)^s v = \mu \quad \text{in } \mathbb{R}^N,$$ ¶ with vanishing conditions at infinity. Here $\mu$ is a bounded Radon measure whose support is compactly contained in $\mathbb{R}^N$, $N\geq2$, and $-(\Delta)^s$ is the fractional Laplace operator of order $s\in (1/2,1)$.

Hitchhiker's guide to the fractional Sobolev spaces

2012

AbstractThis paper deals with the fractional Sobolev spaces Ws,p. We analyze the relations among some of their possible definitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results.Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.

Existence of minimizers for eigenvalues of the Dirichlet-Laplacian with a drift

2015

Abstract This paper deals with the eigenvalue problem for the operator L = − Δ − x ⋅ ∇ with Dirichlet boundary conditions. We are interested in proving the existence of a set minimizing any eigenvalue λ k of L under a suitable measure constraint suggested by the structure of the operator. More precisely we prove that for any c > 0 and k ∈ N the following minimization problem min ⁡ { λ k ( Ω ) : Ω quasi-open set , ∫ Ω e | x | 2 / 2 d x ≤ c } has a solution.

Representation of solutions and large-time behavior for fully nonlocal diffusion equations

2017

Abstract We study the Cauchy problem for a nonlocal heat equation, which is of fractional order both in space and time. We prove four main theorems: (i) a representation formula for classical solutions, (ii) a quantitative decay rate at which the solution tends to the fundamental solution, (iii) optimal L 2 -decay of mild solutions in all dimensions, (iv) L 2 -decay of weak solutions via energy methods. The first result relies on a delicate analysis of the definition of classical solutions. After proving the representation formula we carefully analyze the integral representation to obtain the quantitative decay rates of (ii). Next we use Fourier analysis techniques to obtain the optimal dec…