A parabolic regularization property of p-logarithmic Sobolev generators

Abstract

Let N be a Riemannian manifold, M ⊂ N be a domain with smooth boundary, μ a positive measure on M such that M has unit μ-volume. Consider the evolution driven by the p-Laplace-type operator (p > 2) associated to the natural p-energy functional E (p) con- structed from μ, homogeneous Dirichlet boundary conditions on ∂M being assumed. Assume that a single suitable logarithmic inequality holds for E (p) .Then we show that the evolution brings any data belonging to the natural domain of the evolution instantaneously into L q for any q > 2, with quantitative bounds on the Lq norms. (q > 2 arbitrary) for the evolution equation associated to (possibly degenerate or sin- gular) p-Laplacian-like operators on finite volume domains of Riemannian manifolds, Dirichlet boundary conditions being assumed, provided the associated energy func- tional satisfy a single logarithmic Sobolev inequality. This parallels, in the present case, the results discovered by L. Gross in his cel- ebrated paper (11) for the linear case (see also (12) and, without any claim of com- pleteness, the fundamental papers of Federbush, Nelson, Simon and Hoegh-Krohn(9), (13), (15)), but shows a substantial and unexpected difference with that situation, in which it is well known that no more than a L 2 -L p(t) regularization holds, with p(t) smooth and increasing, p(0 )= 2, p(t) → +∞ as t → +∞. The Ornstein-Uhlenbeck semigroup shows the sharpness of that result in the linear case, this being particularly evident in the fact that the eigenfunctions of such operator are unbounded. To start with we shall introduce our setting and the corresponding notation. We consider a connected, smooth Riemannian manifold (N,g) of dimension n endowed with the associated Riemannian measure m .L etM ⊂ N be an open domain with smooth boundary and consider a measurable function V on M. It will assumed here- after, and will be crucial in what follows, that e V is a probabilitymeasure on M ,s o that the μ-volume of M is one, where we set dμ := e V dm .A ll L p spaces and norms will