A Stefan problem for a protocell model with symmetry-breaking bifurcations of analytic solutions (Q5935832)

The authors consider the following system: NEWLINE\[NEWLINE\begin{aligned} \Delta\mu & =\chi_\Omega \text{ in }\mathbb{R}^2\\ \mu & =\log r\bigl( 1+o(1)\bigr)\text{ as }r= |x|\to \infty\\ \mu & =C^1\text{ across }\partial\Omega\\ -\Delta u & =\mu \text{ in }\Omega\\ u & =0\text{ on }\partial\Omega\\ {\partial u\over \partial n}+\beta & =0\text{ on }\partial \Omega,\;\beta>0.\end{aligned}NEWLINE\]NEWLINE \(\chi_\Omega\) denotes the characteristic function of \(\Omega\). It is a free boundary value problem since \(\Omega\) is not given. The main result is the existence of nonradially symmetric solutions bifurcating from radially symmetric ones.