On the Regularity of Green Functions in Lipschitz Domains

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In this paper we study regularity properties of Green functions associated with elliptic differential operators L in Lipschitz domains. In particular, we discuss the membership of G D (x, ·) and to weak Sobolev spaces in bounded Lipschitz domains Ω, uniformly for x ∈ Ω, where G D is the Green function with Dirichlet boundary condition associated with L, δ∂Ω is the distance function to the boundary of Ω, and α ∈ [0, 1]. Our analysis includes the case of second and higher order elliptic systems with constant coefficients, the bi-Laplacian, as well as the Stokes system.