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The inhomogeneous Dirichlet problems concerning divergence form elliptic equations are studied. Optimal regularity requirements on the coefficients and domains for the W 1, p theory, 1 < p < ∞, are obtained. The principal coefficients are supposed to be in the John-Nirenberg space with small BMO seminorms. The domain is 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.

In this article we prove the regularity of weakly biharmonic maps of domains in Euclidean four space into spheres, as well as the corresponding partial regularity result of stationary biharmonic maps of higher-dimensional domains into spheres.

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