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

Keywords:

• Convexity
• inequalities
• maximum principles