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Abstract
We establish the global Holder estimates for solutions to second-order elliptic equations, which vanish on the boundary, while the right-hand side is allowed to be unbounded. For nondivergence elliptic equations in domains satisfying an cone condition, similar results were obtained by J. H. Michael, who in turn relied on the barrier techniques due to K. Miller. Our approach is based on special growth lemmas, and it works for both divergence and nondivergence, elliptic and parabolic equations, in domains satisfying a general exterior measure condition.
Highlights
In the theory of partial differential equations, it is important to have estimates of solutions, which do not depend on the smoothness of the given data
Such kind of estimates include different versions of the maximum principle, which are crucial for investigation of boundary value problems for second-order elliptic and parabolic equations
In Theorem 3.9, we extend this result to domains Ω satisfying an exterior sphere condition
Summary
In the theory of partial differential equations, it is important to have estimates of solutions, which do not depend on the smoothness of the given data. Such kind of estimates include different versions of the maximum principle, which are crucial for investigation of boundary value problems for second-order elliptic and parabolic equations. Ω is a bounded open set in Rn, n ≥ 1, satisfying the following “exterior measure” condition (A). This condition appeared in the books [4, 5]
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