Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications

Homogenization of a Periodic Degenerate Semilinear Elliptic Pde

In this paper a semilinear elliptic PDE with rapidly oscillating coefficients is homogenized. The novetly of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some portion of Rd. Our fully probabilistic method is based on the connection between PDEs and BSDEs with random terminal time and the weak convergence of a class of diffusion...

Multiscale Young Measures in Almost Periodic Homogenization and Applications

We prove the existence of multiscale Young measures associated with almost periodic homogenization. We give applications of this tool in the homogenization of nonlinear partial differential equations with an almost periodic structure, such as scalar conservation laws, nonlinear transport equations, HamiltonJacobi equations and fully nonlinear elliptic equations. Motivated by the application to ...

Periodic approximations of the ergodic constants in the stochastic homogenization of nonlinear second-order (degenerate) equations

We prove that the effective nonlinearities (ergodic constants) obtained in the stochastic homogenization of Hamilton-Jacobi, “viscous” Hamilton-Jacobi and nonlinear uniformly elliptic pde are approximated by the analogous quantities of appropriate “periodizations” of the equations. We also obtain an error estimate, when there is a rate of convergence for the stochastic homogenization.

Almost Periodic Solutions of First- and Second-Order Cauchy Problems

Almost periodicity of solutions of firstand second-order Cauchy problems on the real line is proved under the assumption that the imaginary (resp. real) spectrum of the underlying operator is countable. Related results have been obtained by Ruess Vu~ and Basit. Our proof uses a new idea. It is based on a factorisation method which also gives a short proof (of the vector-valued version) of Loomi...

Almost Periodic Solutions of Second Order Nonlinear Differential Equations with Almost Periodic Forcing