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

Mahmoudreza Bazarganzadeh Department of Mathematics, Uppsala University Erik Lindgren Department of Mathematics, Norwegian University of Science and Technology

Differential Geometry Metric Geometry mathscidoc: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}.
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@inproceedings{mahmoudreza2012tangential,
  title={Tangential touch between the free and the fixed boundary in a semilinear free boundary problem in two dimensions},
  author={Mahmoudreza Bazarganzadeh, and Erik Lindgren},
  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.
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