Vol. 7, No. 3, 2014

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
A geometric tangential approach to sharp regularity for degenerate evolution equations

Eduardo V. Teixeira and José Miguel Urbano

Vol. 7 (2014), No. 3, 733–744
Abstract

That the weak solutions of degenerate parabolic PDEs modelled on the inhomogeneous p-Laplace equation

ut ÷(|u|p2u) = f Lq,r,p > 2

are C0,α, for some α (0,1), has been known for almost 30 years. What was hitherto missing from the literature was a precise and sharp knowledge of the Hölder exponent α in terms of p,q,r and the space dimension n. We show in this paper that

α = (pq n)r pq q[(p 1)r (p 2)]

using a method based on the notion of geometric tangential equations and the intrinsic scaling of the p-parabolic operator. The proofs are flexible enough to be of use in a number of other nonlinear evolution problems.

Keywords
degenerate parabolic equations, sharp Hölder regularity, tangential equations, intrinsic scaling
Mathematical Subject Classification 2010
Primary: 35K55, 35K65, 35B65
Milestones
Received: 12 August 2013
Revised: 20 January 2014
Accepted: 17 February 2014
Published: 18 June 2014
Authors
Eduardo V. Teixeira
Department of Mathematics
Universidade Federal do Ceará
Campus of Pici
Bloco 914
60455–760 Fortaleza-CE
Brazil
José Miguel Urbano
CMUC, Department of Mathematics
University of Coimbra
3001–501 Coimbra
Portugal