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Differential GeometryMetric Geometrymathscidoc:1701.10013

Arkiv for Matematik, 52, (1), 21-42, 2012.2
We study minimizers of the functionalwhere $B_{1}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .1ex\hbox {$\scriptscriptstyle +$}} {\scriptscriptstyle +}}}=\{x\in B_{1}: x_{1}>0\}$ ,$u$=0 on {$x$∈$B$_{1}:$x$_{1}=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$C$^{1}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 {$x$:±$u$($x$)>0}.
@inproceedings{mahmoudreza2012tangential,
title={Tangential touch between the free and the fixed boundary in a semilinear free boundary problem in two dimensions},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203636647747053},
booktitle={Arkiv for Matematik},
volume={52},
number={1},
pages={21-42},
year={2012},
}

Mahmoudreza Bazarganzadeh, and Erik Lindgren. Tangential touch between the free and the fixed boundary in a semilinear free boundary problem in two dimensions. 2012. Vol. 52. In Arkiv for Matematik. pp.21-42. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203636647747053.