Calculus of Variations and Geometric Measure Theory

P. Baroni - A. Di Castro - G. Palatucci

Global estimates for nonlinear parabolic equations

created by palatucci on 26 Jan 2013
modified by baroni on 31 Aug 2017


Published Paper

Inserted: 26 jan 2013
Last Updated: 31 aug 2017

Journal: J. Evol. Equations
Volume: 13
Number: 1
Pages: 163-195
Year: 2013
Doi: 10.1007/s00028-013-0174-6

ArXiv: 1301.1872 PDF


We consider nonlinear parabolic equations of the type \[ u_t - {\rm div}\, a(x, t, Du)= f(x,t) \qquad \text{on}\qquad \Omega_T = \Omega\times (-T,0), \] under standard growth conditions on $a$, with $f$ only assumed to be integrable. We prove general decay estimates up to the boundary for level sets of the solutions $u$ and the gradient $Du$ which imply very general estimates in Lebesgue and Lorentz spaces. Assuming only that the involved domains satisfy a mild exterior capacity density condition, we provide global regularity results.

Keywords: higher integrability, Calderon-Zygmund theory, Nonlinear parabolic problems, Lorentz regularity