A Polya-Szego inequality in a Fermi metric

作者：McGillivray I^*

来源：Journal of Mathematical Analysis and Applications, 2016, 444(1): 390-432.

DOI：10.1016/j.jmaa.2016.06.033

• Polya-Szego inequality
• Steiner symmetrisation

Let n, k is an element of N with n >= 2 and 1 <= k < n. Given a positive function gamma is an element of C-infinity(Rn-k) we form the Riemannian metric <(g)over tilde> on R-n associated to the differential expression ds(2) = vertical bar dx'vertical bar(2) + gamma(x')(2) vertical bar dy vertical bar(2) where we write R-n (sic) x = (x', y) with x' is an element of Rn-k and y is an element of R-k. Let nu be a log-convex measure on R-k with smooth density and mu the product measure mu := rho Ln-k circle times nu on R-n where rho is an element of C(Rn-k) is a positive function. We obtain a Polya-Szego inequality of the form
integral(Rn)f(u,j(del((g) over tilde)u)) d mu >= integral(Rn)f (u(s), j(del((g) over tilde)u(s)))d mu
for Sobolev functions u where the operation.(s) refers to the (k, n)-Steiner symmetrisation with respect to nu. The gradient operator V-(g) over tilde. is associated to the metric (g) over tilde and the mapping j may be seen as interpolating between the tangent space at x and R-n. The nonnegative integrand f is continuous and convex in the gradient variable and satisfies some additional hypotheses. As an application we derive a Polya-Szego inequality in the hyperbolic plane that takes the above form.

• 出版日期2016-12-1