Abstract
Let n, k ∈ N with n ≥ 2 and 1 ≤ k < n. Given a positive function γ ∈ C∞(Rn−k) we form the Riemannian metric ~g on Rn associated to the differential expression ds2 = |dx'|2 +γ(x')2|dy|2 where we write Rn ∋ x = (x', y) with x' ∈ Rn−k and y ∈ Rk. Let ν be a log-convex measure on Rk with smooth density and µ the product measure µ := ρL n−k⊗ν on Rn where ρ ∈ C(Rn−k)is a positive function. We obtain a Pólya–Szegö inequality of the form
∫f(u, j(∇~gu)) dµ ≥ ∫ f(us, j(∇~gus)) dµ
Rn Rn
for Sobolev functions u where the operation ·s refers to the (k, n)-Steiner symmetrisation withrespect to ν. The gradient operator ∇~g is associated to the metric ~g and the mapping j maybe seen as interpolating between the tangent space at x and Rn. The nonnegative integrand f is continuous and convex in the gradient variable and satisfies some additional hypotheses. As an application we derive a Pólya–Szegö inequality in the hyperbolic plane that takes theabove form
∫f(u, j(∇~gu)) dµ ≥ ∫ f(us, j(∇~gus)) dµ
Rn Rn
for Sobolev functions u where the operation ·s refers to the (k, n)-Steiner symmetrisation withrespect to ν. The gradient operator ∇~g is associated to the metric ~g and the mapping j maybe seen as interpolating between the tangent space at x and Rn. The nonnegative integrand f is continuous and convex in the gradient variable and satisfies some additional hypotheses. As an application we derive a Pólya–Szegö inequality in the hyperbolic plane that takes theabove form
| Original language | English |
|---|---|
| Pages (from-to) | 390-432 |
| Number of pages | 43 |
| Journal | Journal of Mathematical Analysis and Applications |
| Volume | 444 |
| Issue number | 1 |
| Early online date | 28 Jun 2016 |
| DOIs | |
| Publication status | Published - 1 Dec 2016 |
Keywords
- Pólya–Szegö inequality
- Steiner symmetrisation