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 ut=|x|γ∇⋅(|x|−β∇um), with 0<m<1 posed on cylinders of (0,T)×RN. The weights |x|γ and |x|−β, with γ<N and γ−2<β≤γ(N−2)/N can be both degenerate and singular and need not belong to the class A2, 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.