Electron. J. Diff. Eqns., Vol. 2004(2004), No. 44, pp. 1-12.

Boundary monotonicity formulae and applications to free boundary problems I: The elliptic case

Georg S. Weiss

Abstract:
We derive a monotonicity formula at boundary points for a class of nonlinear elliptic partial differential equations, including the obstacle problem case, quenching, a free boundary problem with Bernoulli-type free boundary condition as well as the blow-up case. As application model we prove - for Dirichlet boundary data satisfying certain assumptions - the global existence of a classical solution of the free boundary problem with Bernoulli-type free boundary condition in two and three dimensions.

Submitted October 13, 2003. Published March 29, 2004.
Math Subject Classifications: 35R35, 35J67, 35J60.
Key Words: Free boundary, boundary regularity, non-tangential touch, monotonicity formula, global regularity, Bernoulli-type free boundary condition.

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Georg S. Weiss Graduate School of Mathematical Sciences
University of Tokyo
3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan
email: gw@ms.u-tokyo.ac.jp

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