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Local energy decay for several evolution equations on asymptotically euclidean manifolds

Jean-François Bony, Dietrich Häfner (2012)

Annales scientifiques de l'École Normale Supérieure

Let  P be a long range metric perturbation of the Euclidean Laplacian on  ℝ d , d ≥ 2 . We prove local energy decay for the solutions of the wave, Klein-Gordon and Schrödinger equations associated to  P . The problem is decomposed in a low and high frequency analysis. For the high energy part, we assume a non trapping condition. For low (resp. high) frequencies we obtain a general result about the local energy decay for the group e i t f ( P ) where f has a suitable development at zero (resp. infinity).

Local Exchange Potentials for Electronic Structure Calculations

Eric Cancès, Gabriel Stoltz, Gustavo E. Scuseria, Viktor N. Staroverov, Ernest R. Davidson (2009)

MathematicS In Action

The Hartree-Fock exchange operator is an integral operator arising in the Hartree-Fock model as well as in some instances of the density functional theory. In a number of applications, it is convenient to approximate this integral operator by a multiplication operator, i.e. by a local potential. This article presents a detailed analysis of the mathematical properties of various local approximations to the nonlocal Hartree-Fock exchange operator including the Slater potential, the optimized effective...

Local gradient estimates of p -harmonic functions, 1 / H -flow, and an entropy formula

Brett Kotschwar, Lei Ni (2009)

Annales scientifiques de l'École Normale Supérieure

In the first part of this paper, we prove local interior and boundary gradient estimates for p -harmonic functions on general Riemannian manifolds. With these estimates, following the strategy in recent work of R. Moser, we prove an existence theorem for weak solutions to the level set formulation of the 1 / H (inverse mean curvature) flow for hypersurfaces in ambient manifolds satisfying a sharp volume growth assumption. In the second part of this paper, we consider two parabolic analogues of the p -harmonic...

Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations

Pierre Bousquet (2007)

ESAIM: Control, Optimisation and Calculus of Variations

The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form div a ( ∇ u ) + F [ u ] ( x ) = 0 , over the functions u ∈ W 1 , 1 ( Ω ) that assume given boundary values ϕ on ∂Ω. The vector field a : ℝ n → ℝ n satisfies an ellipticity condition and for a fixed x, F[u](x) denotes a non-linear functional of u. In considering the same problem, Hartman and Stampacchia [Acta Math.115 (1966) 271–310] have obtained existence results in the space of uniformly Lipschitz continuous functions...

Local semiconvexity of Kantorovich potentials on non-compact manifolds

Alessio Figalli, Nicola Gigli (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We prove that any Kantorovich potential for the cost function c = d2/2 on a Riemannian manifold (M, g) is locally semiconvex in the “region of interest”, without any compactness assumption on M, nor any assumption on its curvature. Such a region of interest is of full μ-measure as soon as the starting measure μ does not charge n – 1-dimensional rectifiable sets.

Local semiconvexity of Kantorovich potentials on non-compact manifolds*

Alessio Figalli, Nicola Gigli (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We prove that any Kantorovich potential for the cost function c = d2/2 on a Riemannian manifold (M, g) is locally semiconvex in the “region of interest”, without any compactness assumption on M, nor any assumption on its curvature. Such a region of interest is of full μ-measure as soon as the starting measure μ does not charge n – 1-dimensional rectifiable sets.

Local solvability and regularity results for a class of semilinear elliptic problems in nonsmooth domains

M. Bochniak, Anna-Margarete Sändig (1999)

Mathematica Bohemica

We consider a class of semilinear elliptic problems in two- and three-dimensional domains with conical points. We introduce Sobolev spaces with detached asymptotics generated by the asymptotical behaviour of solutions of corresponding linearized problems near conical boundary points. We show that the corresponding nonlinear operator acting between these spaces is Frechet differentiable. Applying the local invertibility theorem we prove that the solution of the semilinear problem has the same asymptotic...

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