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The Wiener test for degenerate elliptic equations

E. B. Fabes, D. S. Jerison, C. E. Kenig (1982)

Annales de l'institut Fourier

We consider degenerated elliptic equations of the form ∑ i , j D x i ( a i j ( x ) D x j ) , where λ w ( x ) | ξ | 2 ≤ ∑ i , j a i j ( x ) ξ i ξ j ≤ Λ w ( x ) | ξ | 2 . Under suitable assumptions on w , we obtain a characterization of Wiener type (involving weighted capacities) for the set of regular points for these operators. The set of regular points is shown to depend only on w . The main tool we use is an estimate for the Green function in terms of w .

The Wolff gradient bound for degenerate parabolic equations

Tuomo Kuusi, Giuseppe Mingione (2014)

Journal of the European Mathematical Society

The spatial gradient of solutions to non-homogeneous and degenerate parabolic equations of p -Laplacean type can be pointwise estimated by natural Wolff potentials of the right hand side measure.

Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations

Anne de Bouard, Arnaud Debussche, Laurent Di Menza (2001)

Journées équations aux dérivées partielles

We describe several results obtained recently on stochastic nonlinear Schrödinger equations. We show that under suitable smoothness assumptions on the noise, the nonlinear Schrödinger perturbed by an additive or multiplicative noise is well posed under similar assumptions on the nonlinear term as in the deterministic theory. Then, we restrict our attention to the case of a focusing nonlinearity with critical or supercritical exponent. If the noise is additive, smooth in space and non degenerate,...

Theoretical and numerical study of a free boundary problem by boundary integral methods

Michel Crouzeix, Philippe Féat, Francisco-Javier Sayas (2001)

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

In this paper we study a free boundary problem appearing in electromagnetism and its numerical approximation by means of boundary integral methods. Once the problem is written in a equivalent integro-differential form, with the arc parametrization of the boundary as unknown, we analyse it in this new setting. Then we consider Galerkin and collocation methods with trigonometric polynomial and spline curves as approximate solutions.

Theoretical and numerical study of a free boundary problem by boundary integral methods

Michel Crouzeix, Philippe Féat, Francisco-Javier Sayas (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we study a free boundary problem appearing in electromagnetism and its numerical approximation by means of boundary integral methods. Once the problem is written in a equivalent integro-differential form, with the arc parametrization of the boundary as unknown, we analyse it in this new setting. Then we consider Galerkin and collocation methods with trigonometric polynomial and spline curves as approximate solutions.

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