Periodic solutions for nonlinear elliptic equations.
Application to charged particle beam focusing systems

Mihai Bostan; Eric Sonnendrücker

Displaying similar documents to “Periodic solutions for nonlinear elliptic equations. Application to charged particle beam focusing systems”

The Nonlinearly Damped Oscillator

Juan Luis Vázquez (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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We study the large-time behaviour of the nonlinear oscillator $m x^{''} + f (x^{'}) + k x = 0,$ where and is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case $f (x^{'}) = A {| x^{'} |}^{α - 1} x^{'}$ with real, . We characterize the existence and behaviour of fast orbits, , orbits that stop in finite time.

Homogenization and localization in locally periodic transport

Grégoire Allaire, Guillaume Bal, Vincent Siess (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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In this paper, we study the homogenization and localization of a spectral transport equation posed in a locally periodic heterogeneous domain. This equation models the equilibrium of particles interacting with an underlying medium in the presence of a creation mechanism such as, for instance, neutrons in nuclear reactors. The physical coefficients of the domain are -periodic functions modulated by a macroscopic variable, where is a small parameter. The mean free path of the...

Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in (Ω)

M. F. Betta, A. Mercaldo, F. Murat, M. M. Porzio (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is  $\{- div (a (x) (1 + {| \nabla u |}^{2})^{\frac{p - 2}{2}} {\nabla u) + b (x) (1 + | \nabla u |}^{2})^{\frac{λ}{2}} = f in Ω, u = 0 on \partial Ω,$  where Ω is a bounded open subset of $ℝ^{N}$ , N > 2, 2-1/, belongs to (Ω), $a (x) \geq α_{0} > 0$ , is a function in (Ω), is a function in $L^{r} (Ω)$ and 0 ≤ λ < λ *(), for some and λ *().

Sign changing solutions for elliptic equations with critical growth in cylinder type domains

Pedro Girão, Miguel Ramos (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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We prove the existence of positive and of nodal solutions for -Δu = |u|u + µ|u|u, $u \in H_{0}^{1} (Ω)$ , where µ > 0 and 2 < < = , for a class of open subsets of $ℝ^{N}$ lying between two infinite cylinders.

A singular perturbation problem in a system of nonlinear Schrödinger equation occurring in Langmuir turbulence

Cédric Galusinski (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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The aim of this work is to establish, from a mathematical point of view, the limit α → +∞ in the system $i \partial_{t} E + \nabla (\nabla . E) - α^{2} \nabla \times \nabla \times E = - {| E |}^{2 σ} E,$ where $E : ℝ^{3} \to ℂ^{3}$ . This corresponds to an approximation which is made in the context of Langmuir turbulence in plasma Physics. The -subcritical (that is 2/3) and the -subcritical (that is 2) are studied. In the physical case = 1, the limit is then studied for the $H^{1} (ℝ^{3})$ norm.