BIFURCATION STRUCTURE TO SERRIN’S OVERDETERMINED PROBLEM

Abstract

We study the bifurcation structure to Serrin’s overdetermined problem such that

$$-\mathrm{\Delta }u=1\text{ in }\mathrm{\Omega },u=0,{\partial }_{\nu }u=\text{ const on }\partial \mathrm{\Omega }.$$

We prove that the bifurcation from the straight cylinder $B_{{\mathrm{\lambda }}_1}\times ℝ$ with is critical at the bifurcation point. Moreover, we obtain the global structure of bifurcation branches. To study the global structure of bifurcation branches, we establish a global bifurcation theorem in finite dimensional space.