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Stochastic calculus and degenerate boundary value problems

Patrick Cattiaux (1992)

Annales de l'institut Fourier

Consider the boundary value problem (L.P): ( h - A ) u = f in D , ( v - Γ ) u = g on D where A is written as A = 1 / 2 i = 1 m Y i 2 + Y 0 , and Γ is a general Venttsel’s condition (including the oblique derivative condition). We prove existence, uniqueness and smoothness of the solution of (L.P) under the Hörmander’s condition on the Lie brackets of the vector fields Y i ( 0 i m ), for regular open sets D with a non-characteristic boundary.Our study lies on the stochastic representation of u and uses the stochastic calculus of variations for the ( A , Γ ) -diffusion process...

Stochastic homogenization of a class of monotone eigenvalue problems

Nils Svanstedt (2010)

Applications of Mathematics

Stochastic homogenization (with multiple fine scales) is studied for a class of nonlinear monotone eigenvalue problems. More specifically, we are interested in the asymptotic behaviour of a sequence of realizations of the form - div a T 1 x ε 1 ω 1 , T 2 x ε 2 ω 2 , u ε ω = λ ε ω 𝒞 ( u ε ω ) . It is shown, under certain structure assumptions on the random map a ( ω 1 , ω 2 , ξ ) , that the sequence { λ ε ω , k , u ε ω , k } of k th eigenpairs converges to the k th eigenpair { λ k , u k } of the homogenized eigenvalue problem - div ( b ( u ) ) = λ 𝒞 ¯ ( u ) . For the case of p -Laplacian type maps we characterize b explicitly.

Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in 3

M. Burak Erdoğan, Michael Goldberg, Wilhelm Schlag (2008)

Journal of the European Mathematical Society

We present a novel approach for bounding the resolvent of H = - Δ + i ( A · + · A ) + V = : - Δ + L 1 for large energies. It is shown here that there exist a large integer m and a large number λ 0 so that relative to the usual weighted L 2 -norm, ( L ( - Δ + ( λ + i 0 ) ) - 1 ) m < 1 2 2 for all λ > λ 0 . This requires suitable decay and smoothness conditions on A , V . The estimate (2) is trivial when A = 0 , but difficult for large A since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and then sum over...

Strichartz estimates for water waves

Thomas Alazard, Nicolas Burq, Claude Zuily (2011)

Annales scientifiques de l'École Normale Supérieure

In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system with surface tension. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to construct the solutions in [3]. On the other hand, for smoother initial data, we prove that the solutions enjoy the optimal Strichartz estimates (i.e, without loss of regularity compared to the system linearized at ( η = 0 , ψ = 0 )).

Strong convergence estimates for pseudospectral methods

Wilhelm Heinrichs (1992)

Applications of Mathematics

Strong convergence estimates for pseudospectral methods applied to ordinary boundary value problems are derived. The results are also used for a convergence analysis of the Schwarz algorithm (a special domain decomposition technique). Different types of nodes (Chebyshev, Legendre nodes) are examined and compared.

Strong diamagnetism for general domains and application

Soeren Fournais, Bernard Helffer (2007)

Annales de l’institut Fourier

We consider the Neumann Laplacian with constant magnetic field on a regular domain in 2 . Let B be the strength of the magnetic field and let λ 1 ( B ) be the first eigenvalue of this Laplacian. It is proved that B λ 1 ( B ) is monotone increasing for large B . Together with previous results of the authors, this implies the coincidence of all the “third” critical fields for strongly type 2 superconductors.

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