 Abstract : We give a reasonably self-contained exposition of general properties of solutions to second order elliptic and parabolic equations, which do not depend on the regularity of coefficients. These properties include the following.
1. Interior and boundary Hölder regularity of solutions.
2. Interior and boundary Harnack inequalities in Lipschitz and more general domains.
3. Estimates for ratios of solutions and their applications (boundary Hopf-Oleinik estimates, properties of eigenvalues, etc.)
The exposition is based on a few basic facts which have clear probabilistic interpretation, together with elementary geometric constructions. Most of properties under consideration hold true for equations in both non-divergence and divergence forms. |
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