Arkiv för Matematik

Tangential touch between the free and the fixed boundary in a semilinear free boundary problem in two dimensions

Mahmoudreza Bazarganzadeh and Erik Lindgren

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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}.

Article information

Source
Ark. Mat., Volume 52, Number 1 (2014), 21-42.

Dates
Received: 8 February 2012
Revised: 10 September 2012
First available in Project Euclid: 30 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485802654

Digital Object Identifier
doi:10.1007/s11512-012-0179-3

Mathematical Reviews number (MathSciNet)
MR3175292

Zentralblatt MATH identifier
1307.49011

Rights
2013 © Institut Mittag-Leffler

Citation

Bazarganzadeh, Mahmoudreza; Lindgren, Erik. Tangential touch between the free and the fixed boundary in a semilinear free boundary problem in two dimensions. Ark. Mat. 52 (2014), no. 1, 21--42. doi:10.1007/s11512-012-0179-3. https://projecteuclid.org/euclid.afm/1485802654


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