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 ∇u m ), with 0 < m < 1 posed on cylinders of (0, T ) × R N . 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.

On the existence and Hölder regularity of solutions to some nonlinear Cauchy-Neumann problems

“…To the best of our knowledge, this is the first result in this direction: in the existing literature, the results concerning the Hölder regularity of weak solutions to parabolic weighted equations are obtained working in the pure parabolic setting, and are based on the validity of some Harnack inequality in the spirit of Moser [28] (see e.g. [5,10,15,20] and the references therein). We also point out that our method allows us to treat both the existence and the Hölder regularity of weak solutions using the same approximating sequence, in contrast with the classical theory where the two issues are often unrelated.…”

Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results

“…Based on a global Harnack Principle, this result is at the core of the method of [20]. It relies on a quantitative version of the results of J. Moser in [79,80], and a constructive proof of [22] based on the improved results of [21]. As discussed in [20,Chapter 7], the tail decay in (2.12) is not only sufficient but also necessary for obtaining (2.13).…”

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“…The proof in the singular case can be found in [14] and for the treatment of signed solutions we mention [18]. For more recent developments we refer to [3,8,19,20]. However, for the obstacle problem the theory is not complete yet.…”