Multiscale Young Measures in Almost Periodic Homogenization and Applications
Luigi Ambrosio ,
Hermano Frid
Keywords:
homogenization, Young measures, two-scale, almost periodic functions
Abstract:
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 Hamilton-Jacobi equations, fully nonlinear elliptic equations, scalar conservation laws and nonlinear transport equations. Motivated by the application to nonlinear transport equations, we also prove basic results on
flows generated by Lipschitz almost periodic vector fields which are of interest in their own. In our analysis, an important role is played by the so called Bohr compactification ${\mathbb G}^N$ of $\R^N$; this is a natural parameter space for the Young measures. Our homogenization results provide also the asymptotic behaviour for the whole set of ${\mathbb G}^N$-translates of the solutions,
which is in the spirit of recent studies on the homogenization of stationary ergodic processes.
MSC 2000:
35B40 Asymptotic behavior of solutions
35B35 Stability, boundedness
35L65 Conservation laws
35K55 Nonlinear PDE of parabolic type
