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Variable depth KdV equations and generalizations to more nonlinear regimes

Samer Israwi (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We study here the water waves problem for uneven bottoms in a highly nonlinear regime where the small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is known that, for such regimes, a generalization of the KdV equation (somehow linked to the Camassa-Holm equation) can be derived and justified [Constantin and Lannes, Arch. Ration. Mech. Anal. 192 (2009) 165–186] when the bottom is flat. We generalize here this result with a new class of equations taking into account...

Variance decay for functionals of the environment viewed by the particle

Jean-Christophe Mourrat (2011)

Annales de l'I.H.P. Probabilités et statistiques

For the random walk among random conductances, we prove that the environment viewed by the particle converges to equilibrium polynomially fast in the variance sense, our main hypothesis being that the conductances are bounded away from zero. The basis of our method is the establishment of a Nash inequality, followed either by a comparison with the simple random walk or by a more direct analysis based on a martingale decomposition. As an example of application, we show that under certain conditions,...

Variational analysis for a nonlinear elliptic problem on the Sierpiński gasket

Gabriele Bonanno, Giovanni Molica Bisci, Vicenţiu Rădulescu (2012)

ESAIM: Control, Optimisation and Calculus of Variations

Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear term, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem on the Sierpiński gasket is proved. Our approach is based on variational methods and on some analytic and geometrical properties of the Sierpiński fractal. The abstract results are illustrated by explicit examples.

Variational inequalities and rearrangements

Angelo Alvino, Silvano Matarasso, Guido Trombetti (1992)

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

We give comparison results for solutions of variational inequalities, related to general elliptic second order operators, involving solutions of symmetrized problems, using Schwarz spherical symmetrization.

Various kinds of sensitive singular perturbations

Nicolas Meunier, Jacqueline Sanchez-Hubert, Évariste Sanchez-Palencia (2007)

Annales mathématiques Blaise Pascal

We consider variational problems of P. D. E. depending on a small parameter ε when the limit process ε ↓ 0 implies vanishing of the higher order terms. The perturbation problem is said to be sensitive when the energy space of the limit problem is out of the distribution space, so that the limit problem is out of classical theory of P. D. E. We present here a review of the subject, including abstract convergence theorems and two very different model problems (the second one is presented for the first...

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