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

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 {x∈B₁: 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 {x:±u(x)>0}.