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A weighted symmetrization for nonlinear elliptic and parabolic equations in inhomogeneous media

Guillermo Reyes, Juan Luis Vázquez (2006)

Journal of the European Mathematical Society

In the theory of elliptic equations, the technique of Schwarz symmetrization is one of the tools used to obtain a priori bounds for classical and weak solutions in terms of general information on the data. A basic result says that, in the absence of lower-order terms, the symmetric rearrangement of the solution u of an elliptic equation, that we write u * , can be compared pointwise with the solution of the symmetrized problem. The main question we address here is the modification of the method to...

A well-posedness result for a mass conserved Allen-Cahn equation with nonlinear diffusion

Kettani, Perla El, Hilhorst, Danielle, Lee, Kai (2017)

Proceedings of Equadiff 14

In this paper, we prove the existence and uniqueness of the solution of the initial boundary value problem for a stochastic mass conserved Allen-Cahn equation with nonlinear diffusion together with a homogeneous Neumann boundary condition in an open bounded domain of n with a smooth boundary. We suppose that the additive noise is induced by a Q-Brownian motion.

A WKB analysis of the Alfvén spectrum of the linearized magnetohydrodynamics equations

Manuel Núñez, Jesús Rojo (1993)

Applications of Mathematics

Small perturbations of an equilibrium plasma satisfy the linearized magnetohydrodynamics equations. These form a mixed elliptic-hyperbolic system that in a straight-field geometry and for a fixed time frequency may be reduced to a single scalar equation div A 1 Δ u + A 2 u = 0 , where A 1 may have singularities in the domaind U of definition. We study the case when U is a half-plane and u possesses high Fourier components, analyzing the changes brought about by the singularity A 1 = . We show that absorptions of energy takes...

Abbildungen harmonischer Raüme mit Anwendung auf die Laplace und Wärmeleitungsgleichung

Wolfhard Hansen (1971)

Annales de l'institut Fourier

This paper is devoted to a study of harmonic mappings φ of a harmonic space E ˜ on a harmonic space E which are related to a family of harmonic mappings of E ˜ into E ˜ . In this way balayage in E may be reduced to balayage in E . In particular, a subset A of E is polar if and only if φ - 1 ( A ) is polar. Similar result for thinness. These considerations are applied to the heat equation and the Laplace equation.

About a Pólya-Schiffer inequality

Bodo Dittmar, Maren Hantke (2011)

Annales UMCS, Mathematica

For simply connected planar domains with the maximal conformal radius 1 it was proven in 1954 by G. Pólya and M. Schiffer that for the eigenvalues λ of the fixed membrane for any n the following inequality holds [...] where λ(o) are the eigenvalues of the unit disk. The aim of the paper is to give a sharper version of this inequality and for the sum of all reciprocals to derive formulas which allow in some cases to calculate exactly this sum.

About a Variant of the 1 d Vlasov equation, dubbed “Vlasov-Dirac-Benney Equation"

Claude Bardos (2012/2013)

Séminaire Laurent Schwartz — EDP et applications

This is a report on project initiated with Anne Nouri [3], presently in progress, with the collaboration of Nicolas Besse [2] ([2] is mainly the material of this report) . It concerns a version of the Vlasov equation where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the full non linear problem and also on natural connections with several other equations of mathematical physic.

About asymptotic approximations in thin waveguides

Nicole Turbe, Louis Ratier (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We study the propagation of electromagnetic waves in a guide the section of which is a thin annulus. Owing to the presence of a small parameter, explicit approximations of the TM and TE eigenmodes are obtained. The cases of smooth and non smooth boundaries are presented.

About asymptotic approximations in thin waveguides

Nicole Turbe, Louis Ratier (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We study the propagation of electromagnetic waves in a guide the section of which is a thin annulus. Owing to the presence of a small parameter, explicit approximations of the TM and TE eigenmodes are obtained. The cases of smooth and non smooth boundaries are presented.

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 Delaunay triangulations and discrete maximum principles for the linear conforming FEM applied to the Poisson equation

Reiner Vanselow (2001)

Applications of Mathematics

The starting point of the analysis in this paper is the following situation: “In a bounded domain in 2 , let a finite set of points be given. A triangulation of that domain has to be found, whose vertices are the given points and which is ‘suitable’ for the linear conforming Finite Element Method (FEM).” The result of this paper is that for the discrete Poisson equation and under some weak additional assumptions, only the use of Delaunay triangulations preserves the maximum principle.

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...

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