Extremal domains in ${\mathbb{S}}^2$: Geometric and Analytic methods

José M. Espinar, Diego A. Marín

In this article, we study domains $\Omega \subset {\mathbb{S}}^2$ that support positive solutions of the overdetermined problem $$\Delta u+f(u,|\nabla u|)=0\text{in }\Omega ,$$ subject to…