Capacitary estimates for solutions of the Dirichlet problem for second order elliptic equations in divergence form (Q1973267)

The Dirichlet problem for the uniformly elliptic equation in divergence form \( \text{div} ( {\mathcal A}(x) \nabla u(x)) \) is considered in an open subset of \( \Omega \subset \mathbb{R}^{n} \). The coefficients of the equation are assumed to be bounded and measurable, whereas no assumption is made on the boundary of the open set \( \Omega \). A pointwise estimate of the solution is derived, improving previously existent estimates. As a particular case one can derive a new sufficient but not necessary condition for Hölder continuity at a boundary point of \( {\mathcal A} \)-harmonic functions vanishing on part of the boundary.