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A theorem of real analysis and its application to free boundary problems

✍ Scribed by Ioannis Athanasopoulos; Luis A. Caffarelli

Publisher: John Wiley and Sons
Year: 1985
Tongue: English
Weight: 148 KB
Volume: 38
Category: Article
ISSN: 0010-3640
DOI: 10.1002/cpa.3160380503

No coin nor oath required. For personal study only.

✦ Synopsis

In this note we show how comparison theorems for solutions to equations with non-smooth coefficients yield in a .very simple fashion some free boundary regularity theorems in cases where there is Q priori geometrical information, for instance, when one knows that the solution, I(, is monotone in some given direction.

1. Statement of the Problems

We shall consider two classical problems. (a) The filtration (or obstacle) problem, where we are given a local minimizer

U, of the energy integral (A > 0) on K , = { u : ~€ H ~, O B , , U -I ( E H ~( B ~)

(with the same techniques one may also consider its quasilinear version, the membrane problem, where the Dirichlet integral is replaced by the area integral).

(b) The Signorini problem, where we are given a local minimizer U, of

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A theorem of real analysis and its application to free boundary problems

✦ Synopsis

1. Statement of the Problems

U, of the energy integral (A > 0) on K , = { u : ~€ H ~, ~~O ~~B , , U -I ( E H ~( B ~)

U, of the energy integral (A > 0) on K , = { u : ~€ H ~, O B , , U -I ( E H ~( B ~)