Quantitative a Priori Estimates for Fast Diffusion Equations with Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and Hölder continuity

Abstract

We study a priori estimates for a class of non-negative local weak solution to the weighted fast diffusion equation $u_t=|x{|}^{\gamma }\nabla \cdot (|x{|}^{-\beta }\nabla u^m)$, with $0<m<1$ posed on cylinders of $(0,T)\times {\mathbb{R}}^N$. The weights $|x{|}^{\gamma }$ and $|x{|}^{-\beta }$, with $\gamma <N$ and $\gamma -2<\beta \le \gamma (N-2)/N$ can be both degenerate and singular and need not belong to the class ${\mathcal{A}}_2$, a typical assumption for this kind of problems. This range of parameters is optimal for the validity of a class of Caffarelli-Kohn-Nirenberg inequalities, which play the role of the standard Sobolev inequalities in this more complicated weighted setting. The weights that we consider are not translation invariant and this causes a number of extra difficulties and a variety of scenarios: for instance, the scaling properties of the equation change when considering the problem around the origin or far from it. We therefore prove quantitative - with computable constants - upper and lower estimates for local weak solutions, focussing our attention where a change of geometry appears. Such estimates fairly combine into forms of Harnack inequalities of forward, backward and elliptic type. As a consequence, we obtain Hölder continuity of the solutions, with a quantitative (even if non-optimal) exponent. Our results apply to a quite large variety of solutions and problems. The proof of the positivity estimates requires a new method and represents the main technical novelty of this paper. Our techniques are flexible and can be adapted to more general settings, for instance to a wider class of weights or to similar problems posed on Riemannian manifolds, possibly with unbounded curvature. In the linear case, $m=1$, we also prove quantitative estimates, recovering known results in some cases and extending such results to a wider class of weights.

53 pages, 2 figures. This is a much improved version of the older manuscript, thanks to the suggestions of the anonymous referee