Homogenization Theory for Partial Differential Equations with Large, Random Potential.

摘要

Partial differential equations with highly oscillatory, random coefficients describe many applications in applied science and engineering such as porous media and composite materials. Homogenization of PDE states that the solution of the initial model converges to the solution to a macro model, which is characterized by the PDE with homogenized coefficients. Particularly, we study PDEs with a large potential, a class of PDEs with a potential properly scaled such that the limiting equation has a non-trivial (non-zero) potential.;This thesis consists of the investigation of three issues. The first issue is the convergence of Schödinger equation to a deterministic homogenized PDE in high dimension. The second issue is the convergence of the same equation to a stochastic PDE in low dimension. The third issue is the convergence of elliptic equation with an imaginary potential.