Least Supersolution Approach to Regularizing Free Boundary Problems |
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Authors: | Diego R Moreira |
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Institution: | (1) Department of Mathematics, University of Texas at Austin, RLM 12.128, Austin, TX 78712-1082, USA;(2) Present address: Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419, USA |
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Abstract: | In this paper, we study a free boundary problem obtained as a limit as ε → 0 to the following regularizing family of semilinear equations , where β
ε
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 in ε. This allows to prove that the free boundary of a limit has the “right” weak geometry, in the measure theoretical sense.
By the construction of some barriers with curvature, the classification of global profiles of the blow-up analysis is carried
out and the limit functions are proven to be viscosity and pointwise solution ( almost everywhere) to a free boundary problem. Finally, the free boundary is proven to be a C
1,α surface around almost everywhere point.
An erratum to this article can be found at |
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Keywords: | |
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