Hölder regularity of solutions to second-order elliptic equations in nonsmooth domains.

Cho, Sungwon; Safonov, Mikhail

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Let $X = (X_{1}, X_{2}, \dots, X_{q})$ be a system of vector fields satisfying the Hörmander condition. We prove $L_{X}^{2, λ}$ local regularity for the gradient $X u$ of a solution of the following strongly elliptic system $- X_{α}^{*} (a_{i j}^{α β} (x) X_{β} u^{j}) = g_{i} - X_{α}^{*} f_{i}^{α} (x) \forall i = 1, 2, \dots, N,$ where $a_{i j}^{α β} (x)$ are bounded functions and belong to Vanishing Mean Oscillation space.

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