Indexado en

Licencia y uso

Citaciones

Cited 16 times in

Cited 11 times in

Cited 23 times in

Altmetrics

Grant support

This work was partially funded by Projects MTM2014-52240-P and MTM2017-85757-P (Spanish Government), and by the E.U. H2020 MSCA programme, grant agreement 777822. N.S. was partially funded by the FPI-grant BES-2015-072962, associated to the project MTM2014-52240-P (Spain). We would like to warmly thank the anonymous reviewer for an extraordinary work that helped us a lot to improve the paper to its current status.

Análisis de autorías institucional

Compartir

Quantitative a priori estimates for fast diffusion equations with Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and Holder continuity

Publicado en:ADVANCES IN MATHEMATICS. 345 1075-1161 - 2019-03-17 345(), DOI: 10.1016/j.aim.2019.01.018

Autores: Bonforte, Matteo; Simonov, Nikita;

Afiliaciones

Resumen

We study a priori estimates for a class of non-negative local weak solution to the weighted fast diffusion equation u(t) = vertical bar x vertical bar(gamma)del . (vertical bar x vertical bar(-beta)del u(m)), with 0 < m < 1 posed on cylinders of (0, T) x R-N. The weights vertical bar x vertical bar(gamma) and vertical bar x vertical bar(-beta), with gamma < N and gamma - 2 < beta <= gamma(N - 2)/N can be both degenerate and singular and need not belong to the class 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 Holder 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. (C) 2019 Elsevier Inc. All rights reserved.

Palabras clave