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On the Newton-Kantorovich theorem and nonlinear finite element methods

Ioannis K. Argyros (2009)

Applicationes Mathematicae

Using a weaker version of the Newton-Kantorovich theorem, we provide a discretization result to find finite element solutions of elliptic boundary value problems. Our hypotheses are weaker and under the same computational cost lead to finer estimates on the distances involved and a more precise information on the location of the solution than before.

On the nodal set of the second eigenfunction of the laplacian in symmetric domains in R N

Lucio Damascelli (2000)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We present a simple proof of the fact that if Ω is a bounded domain in R N , N 2 , which is convex and symmetric with respect to k orthogonal directions, 1 k N , then the nodal sets of the eigenfunctions of the laplacian corresponding to the eigenvalues λ 2 , , λ k + 1 must intersect the boundary. This result was proved by Payne in the case N = 2 for the second eigenfunction, and by other authors in the case of convex domains in the plane, again for the second eigenfunction.

On the nonhamiltonian character of shocks in 2-D pressureless gas

Yu. G. Rykov (2002)

Bollettino dell'Unione Matematica Italiana

The paper deals with the 2-D system of gas dynamics without pressure which was introduced in 1970 by Ua. Zeldovich to describe the formation of largescale structure of the Universe. Such system occurs to be an intermediate object between the systems of ordinary differential equations and hyperbolic systems of PDE. The main its feature is the arising of singularities: discontinuities for velocity and d-functions of various types for density. The rigorous notion of generalized solutions in terms of...

On the nonlinear Neumann problem at resonance with critical Sobolev nonlinearity

J. Chabrowski, Shusen Yan (2002)

Colloquium Mathematicae

We consider the Neumann problem for the equation - Δ u - λ u = Q ( x ) | u | 2 * - 2 u , u ∈ H¹(Ω), where Q is a positive and continuous coefficient on Ω̅ and λ is a parameter between two consecutive eigenvalues λ k - 1 and λ k . Applying a min-max principle based on topological linking we prove the existence of a solution.

On the nonlinear stabilization of the wave equation

Aissa Guesmia (1998)

Annales Polonici Mathematici

We obtain a precise decay estimate of the energy of the solutions to the initial boundary value problem for the wave equation with nonlinear internal and boundary feedbacks. We show that a judicious choice of the feedbacks leads to fast energy decay.

On the number of positive solutions of singularly perturbed 1D nonlinear Schrödinger equations

Patricio Felmer, Salomé Martínez, Kazunaga Tanaka (2006)

Journal of the European Mathematical Society

We study singularly perturbed 1D nonlinear Schrödinger equations (1.1). When V ( x ) has multiple critical points, (1.1) has a wide variety of positive solutions for small ε and the number of positive solutions increases to as ε 0 . We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of V ( x ) . Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.

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