06-366 Diego R. Moreira

Least Supersolution Approach to Regularizing Free Boundary Problems. (610K, PDF) Dec 18, 06

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Abstract. In this paper, we study a free boundary problem obtained as a limit as $\varepsilon \to 0$ to the following regularizing family of semilinear equations $\Delta u = \beta_{\varepsilon}(u) F(\nabla u)$, where $\beta_{\varepsilon}$ approximates the Dirac delta in the origin and $F$ is a Lipschitz function bounded away from $0$ and infinity. The least supersolution approach is used to construct solutions satisfying geometric properties of the level surfaces that are uniform. This allows to prove that the free boundary of the limit has the "right" weak geometry, in the measure theoretical sense. By the construction of some barriers with curvature, the classification of global profiles for the blow-up analysis is carried out and the limit function is proven to be a viscosity and pointwise solution (a.e) to a free boundary problem. Finally, the free boundary is proven to be a $C^{1,\alpha}$ surface around $\mathcal{H}^{n-1}$ a.e. point.

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