A Harnack Inequality Approach to the Regularity of Free Boundaries. Part I: Lipschitz Free Boundaries are $C^{1, \alpha}$ 论文

摘要

This is the first in a series of papers where we intend to show, in several steps, the existence of classical (or as classical as possible) solutions to a general two-phase free-boundary system. We plan to do so by: (a) constructing rather weak generalized solutions of the free-boundary problems, (b) showing that the free boundary of such solutions have nice measure theoretical properties (i.e., finite (n-1)-dimensional Hausdorff measure and the associated differentiability properties), (c) showing that near a flat point free-boundaries are Lipschitz graphs, and (d) showing that Lipschitz free boundaries are really C1,a.