Global gradient estimates for degenerate parabolic equations in nonsmooth domains

Abstract

This paper studies the global regularity theory for degenerate nonlinear parabolic partial differential equations. Our objective is to show that weak solutions belong to a higher Sobolev space than assumed a priori if the complement of the domain satisfies a capacity density condition and if the boundary values are sufficiently smooth. Moreover, we derive integrability estimates for the gradient. The results extend to the parabolic systems as well. The higher integrability estimates provide a useful tool in several applications.

Self-improving property of nonlinear higher order parabolic systems near the boundary

“…The next lemma slightly extends the capacity estimate from the above definition (cf. [36], Lemma 3.8).…”

“…For the proof, see Chapter 10 of Maz'ja's monograph [32] or Hedberg [23] and also [36]. Later we combine this estimate with the boundary regularity condition and obtain a boundary version of Sobolev's inequality.…”

“…The idea to consider parabolic cylinders whose scaling depends on the solution itself goes back to DiBenedetto and Friedman [12,13,14]. The local higher integrability result was recently extended to higher order parabolic systems in [9], and global higher integrability results for quasiminimizers and second order parabolic systems were obtained in [35,36,37].…”

“…Choosing η > 0 small enough and employing a suitable version of (6.4) as in [36] for h, we can absorb the first term on the right-hand side into the left, because the left-hand side matches with (6.4). Moreover, with the help of Hölder's inequality and (6.5), multiplied by α −p , we deduce…”

“…Therefore, treating the remaining terms in (6.11) in a similar way (see also [36], proof of Theorem 4.7) we end up with…”