Convexity properties for solutions of some second order elliptic semilinear equations

Abstract

We investigate the equation div[expH(|Du|2)Du]=0 in a domain ω⊂IR N where H(t) is a function satisfying l+2tH′(t) >0.We derive a maximum principle for IDul f(u), where f(t) is a suitable function. In the case in which ω is a topological ring with boundary Γ0∪Γ1, and u = 0 on Γ0, u =1 on Γ1, we obtain some convexity properties for |Du|. If H(t) = (p/2-l)logt, 1<p (p-capacity problem) we solve a free boundary problem and prove some isoperimetric inequalities