Poincaré inequalities and Steiner symmetrization

A complete geometric characterization for a general Steiner symmetric domain Ω ⊂ Rn to satisfy the Poincare inequality with exponent p > n−1 is obtained and it is shown that this range of exponents is best possible. In the case where the Steiner symmetric domain is determined by revolving the graph of a Lipschitz continuous function, it is shown that the preceding characterization works for all p > 1 and furthermore for such domains a geometric characterization for a more general Sobolev–Poincare inequality to hold is given. Although the operation of Steiner symmetrization need not always preserve a Poincare inequality, a general class of domains is given for which Poincare inequalities are…