Optimal W1, p regularity theory of elliptic and parabolic equations.

摘要

The inhomogeneous Dirichlet problems concerning divergence form elliptic and parabolic equations are studied. The corresponding Neumann problems are also investigated. Optimal regularity requirements on the coefficients and domains for the W1,p theory are obtained. The principal coefficients are supposed to be in the John-Nirenberg space with small BMO semi-norms. The domain is either a Lipschitz domain or a Reifenberg domain. These conditions for the W 1,p theory not only weaken the requirements on the coefficients but also lead to a more general geometric condition on the domains. In fact, these domains might have fractal dimensions.