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April 2014 Tangential touch between the free and the fixed boundary in a semilinear free boundary problem in two dimensions
Mahmoudreza Bazarganzadeh, Erik Lindgren
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Ark. Mat. 52(1): 21-42 (April 2014). DOI: 10.1007/s11512-012-0179-3

Abstract

We study minimizers of the functional $$\int_{B^+_1}(|\nabla u|^2 + 2(\lambda^+(u^+)^p+\lambda^-(u^-)^p))dx, $$where $B_{1}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .1ex\hbox {$\scriptscriptstyle +$}} {\scriptscriptstyle +}}}=\{x\in B_{1}: x_{1}>0\}$, u=0 on {xB1: x1=0}, $\lambda^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle \pm $}} {\raise .17ex\hbox {$\scriptstyle \pm $}} {\raise .1ex\hbox {$\scriptscriptstyle \pm $}} {\scriptscriptstyle \pm }}}$ are two positive constants and 0< p<1. In two dimensions, we prove that the free boundary is a uniform C1 graph up to the flat part of the fixed boundary and also that two-phase points cannot occur on this part of the fixed boundary. Here, the free boundary refers to the union of the boundaries of the sets {xu(x)>0}.

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Mahmoudreza Bazarganzadeh. Erik Lindgren. "Tangential touch between the free and the fixed boundary in a semilinear free boundary problem in two dimensions." Ark. Mat. 52 (1) 21 - 42, April 2014. https://doi.org/10.1007/s11512-012-0179-3

Information

Received: 8 February 2012; Revised: 10 September 2012; Published: April 2014
First available in Project Euclid: 30 January 2017

zbMATH: 1307.49011
MathSciNet: MR3175292
Digital Object Identifier: 10.1007/s11512-012-0179-3

Rights: 2013 © Institut Mittag-Leffler

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Vol.52 • No. 1 • April 2014
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