Serrin type problem without boundary assumption.

Problem's Description

Let $D$ be a bounded open domain and assume for some $u\in W_0^{1,2}(D)$ and $\alpha>0$ the following is true

(i) $-\Delta u=\text{sgn} (u-\alpha)$ in $D$,

(ii) $\int_D \text{sgn} (u-\alpha)dx=0$,

(iii) $u>0$ in $D$.

Show that $D$ is a ball.

1. Article A symmetry problem in potential theory

   • James Serrin

   Archive for Rational Mechanics and Analysis, 1971