Displaying similar documents to “Upper bounds for a class of energies containing a non-local term”

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

Mihai Bostan, Eric Sonnendrücker (2006)

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

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We study the existence of spatial periodic solutions for nonlinear elliptic equations - Δ u + g ( x , u ( x ) ) = 0 , x ∈ ℝ N where g is a continuous function, nondecreasing w.r.t. u . We give necessary and sufficient conditions for the existence of periodic solutions. Some cases with nonincreasing functions g are investigated as well. As an application we analyze the mathematical model of electron beam focusing system and we prove the existence of positive periodic solutions for the envelope equation. We present also numerical...

Non-local approximation of free-discontinuity problems with linear growth

Luca Lussardi, Enrico Vitali (2007)

ESAIM: Control, Optimisation and Calculus of Variations

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We approximate, in the sense of -convergence, free-discontinuity functionals with linear growth in the gradient by a sequence of non-local integral functionals depending on the average of the gradients on small balls. The result extends to higher dimension what we already proved in the one-dimensional case.

A global differentiability result for solutions of nonlinear elliptic problems with controlled growths

Luisa Fattorusso (2008)

Czechoslovak Mathematical Journal

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Let Ω be a bounded open subset of ℝ n , n > 2 . In Ω we deduce the global differentiability result u ∈ H 2 ( Ω , ℝ N ) for the solutions u ∈ H 1 ( Ω , ℝ n ) of the Dirichlet problem u - g ∈ H 0 1 ( Ω , ℝ N ) , - ∑ i D i a i ( x , u , D u ) = B 0 ( x , u , D u ) with controlled growth and nonlinearity q = 2 . The result was obtained by first extending the interior differentiability result near the boundary and then proving the global differentiability result making use of a covering procedure.