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A variational analysis of a gauged nonlinear Schrödinger equation

Alessio Pomponio, David Ruiz (2015)

Journal of the European Mathematical Society

This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: - Δ u ( x ) + ω + h 2 ( | x | ) | x | 2 + ∫ | x | + ∞ h ( s ) s u 2 ( s ) d s u ( x ) = | u ( x ) | p - 1 u ( x ) , where h ( r ) = 1 2 ∫ 0 r s u 2 ( s ) d s . This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for p ∈ ( 1 , 3 ) , the functional may be bounded from below or not, depending on ω . Quite surprisingly, the threshold value for ω is explicit. From...

About boundary terms in higher order theories

Lorenzo Fatibene, Mauro Francaviglia, S. Mercadante (2011)

Communications in Mathematics

It is shown that when in a higher order variational principle one fixes fields at the boundary leaving the field derivatives unconstrained, then the variational principle (in particular the solution space) is not invariant with respect to the addition of boundary terms to the action, as it happens instead when the correct procedure is applied. Examples are considered to show how leaving derivatives of fields unconstrained affects the physical interpretation of the model. This is justified in particular...

About global existence and asymptotic behavior for two dimensional gravity water waves

Thomas Alazard (2012/2013)

Séminaire Laurent Schwartz — EDP et applications

The main result of this talk is a global existence theorem for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description in physical coordinates of the solution, which shows that modified scattering holds.The proof is based on a bootstrap argument involving L 2 and L ∞ estimates. The L 2 bounds are proved in the paper [5]. They rely on a normal forms paradifferential method allowing one to obtain energy estimates on the Eulerian formulation...

About stability and regularization of ill-posed elliptic Cauchy problems: the case of C1,1 domains

Laurent Bourgeois (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with C1,1 boundary. It is an extension of an earlier result of [Phung, ESAIM: COCV9 (2003) 621–635] for domains of class C∞. Our estimate is established by using a Carleman estimate near the boundary in which the exponential weight depends on the distance function to the boundary. Furthermore, we prove that this stability estimate is nearly optimal and induces...

Abstract version of the Cauchy-Kowalewski problem

Oleg Zubelevich (2004)

Open Mathematics

We consider an abstract version of the Cauchy-Kowalewski Problem with the right hand side being free from the Lipschitz type conditions and prove the existence theorem.

Affine ultraregular generalized functions

Khaled Benmeriem, Chikh Bouzar (2010)

Banach Center Publications

Algebras of ultradifferentiable generalized functions satisfying some regularity assumptions are introduced. We give a microlocal analysis within these algebras related to the affine regularity type and the ultradifferentiability property. As a particular case we obtain new algebras of Gevrey generalized functions.

Currently displaying 81 – 100 of 1606