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Analysis of a free boundary at contact points with Lipschitz data


Authors: A. L. Karakhanyan and H. Shahgholian
Journal: Trans. Amer. Math. Soc. 367 (2015), 5141-5175
MSC (2010): Primary 35R35
DOI: https://doi.org/10.1090/S0002-9947-2015-06187-X
Published electronically: March 4, 2015
MathSciNet review: 3335413
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Abstract: In this paper we consider a minimization problem for the functional

$\displaystyle J(u)=\int _{B_1^+}\vert\nabla u\vert\sp 2+\lambda _{+}^2\chi _{\{u>0\}}+\lambda _{-}^2\chi _{\{u\leq 0\}} $

in the upper half ball $ B_1^+\subset \mathbb{R}^n, n\geq 2$, subject to a Lipschitz continuous Dirichlet data on $ \partial B_1^+$. More precisely we assume that $ 0\in \partial \{u>0\}$ and the derivative of the boundary data has a jump discontinuity. If $ 0\in \overline {\partial ( \{u>0\} \cap B_1^+)}$, then (for $ n=2$ or $ n\geq 3$ and the one-phase case) we prove, among other things, that the free boundary $ \partial \{u>0\}$ approaches the origin along one of the two possible planes given by

$\displaystyle \gamma x_1 = \pm x_2, $

where $ \gamma $ is an explicit constant given by the boundary data and $ \lambda _\pm $ the constants seen in the definition of $ J(u)$. Moreover the speed of the approach to $ \gamma x_1=x_2$ is uniform.

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Additional Information

A. L. Karakhanyan
Affiliation: Maxwell Institute for Mathematical Sciences and School of Mathematics, University of Edinburgh, King’s Buildings, Mayfield Road, EH9 3JZ, Edinburgh, Scotland
Email: aram.karakhanyan@ed.ac.uk

H. Shahgholian
Affiliation: Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden
Email: henriksh@math.kth.se

DOI: https://doi.org/10.1090/S0002-9947-2015-06187-X
Keywords: Free boundary problem, regularity, contact points
Received by editor(s): May 22, 2012
Received by editor(s) in revised form: May 15, 2013
Published electronically: March 4, 2015
Additional Notes: The second author was partially supported by the Swedish Research Council. The authors also thank Professor Carlos Kenig for several valuable comments. The first author thanks the Göran Gustafsson Foundation for visiting appointments to KTH
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.