A weighted isoperimetric inequality and applications to symmetrization. (English) Zbl 1029.26018

Betta, M. F.; Brock, F.; Mercaldo, A.; Possteraro, M. R.

J. Inequal. Appl. 4, No. 3, 215-240 (1999).

Let \(a: [0, \infty) \rightarrow [0, \infty)\) be an increasing function satisfying a certain convexity condition. For bounded Lipschitz domains \(\Omega \subset {\mathbb R}^n,\) the authors prove that \(\int_{\partial \Omega}a(|x|)H_{n-1}(dx)\) does not increase when \(\Omega\) is replaced by the ball of the same Lebesgue measure centered at the origin. For \(a \equiv 1,\) this is the standard isoperimetric inequality in \({\mathbb R}^n.\) From this result, the authors go on to prove weighted versions involving the function \(a(|x|)\) that Schwarz symmetrization (i.e., passage to symmetric decreasing rearrangement) decreases Dirichlet integrals and perimeters of Caccioppoli sets. They prove also some comparison theorems of Talenti-type for solutions to some degenerate elliptic p.d.e’s.

Reviewer: Albert Baernstein (St.Louis)

MSC:

26D20	Other analytical inequalities
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J70	Degenerate elliptic equations

Keywords:

symmetrization; isoperimetric inequality