80 research outputs found
Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments
We consider the homogenization of Hamilton-Jacobi equations and degenerate
Bellman equations in stationary, ergodic, unbounded environments. We prove
that, as the microscopic scale tends to zero, the equation averages to a
deterministic Hamilton-Jacobi equation and study some properties of the
effective Hamiltonian. We discover a connection between the effective
Hamiltonian and an eikonal-type equation in exterior domains. In particular, we
obtain a new formula for the effective Hamiltonian. To prove the results we
introduce a new strategy to obtain almost sure homogenization, completing a
program proposed by Lions and Souganidis that previously yielded homogenization
in probability. The class of problems we study is strongly motivated by
Sznitman's study of the quenched large deviations of Brownian motion
interacting with a Poissonian potential, but applies to a general class of
problems which are not amenable to probabilistic tools.Comment: 51 pages, 2 figures. We have added material and made some corrections
to our previous versio
Phasefield theory for fractional diffusion-reaction equations and applications
This paper is concerned with diffusion-reaction equations where the classical
diffusion term, such as the Laplacian operator, is replaced with a singular
integral term, such as the fractional Laplacian operator. As far as the
reaction term is concerned, we consider bistable non-linearities. After
properly rescaling (in time and space) these integro-differential evolution
equations, we show that the limits of their solutions as the scaling parameter
goes to zero exhibit interfaces moving by anisotropic mean curvature. The
singularity and the unbounded support of the potential at stake are both the
novelty and the challenging difficulty of this work.Comment: 41 page
An introduction to the theory of homogenization for first and second order elliptic pde in random environments
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