Boundary estimates for certain degenerate and singular parabolic equations

Abstract

We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic p-Laplacian. Assuming that such solutions continuously vanish on some distinguished part of the lateral part ST of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations, of porous medium type.

On time dependent domains for the degenerate $$p$$ p -parabolic equation: Carleson estimate and Hölder continuity

“…Remark 1 As was seen in [1], that if E ⊂ R N is an NTA-domain as defined in [14] and T > 0, then for any point…”

“…If we restrict ourselves to domains with the boundary given by a time varying graph. That is, let (1) and require that the regularity of A is in a Hölder class for some exponent. We see by a theorem of Petrowsky [21] (or [8]) that if A satisfies a Lipschitz condition in space and Hölder with exponent 1/2 in time then the boundary is regular.…”

“…these mixed type Lipschitz domains are the only Hölder class which is scaling invariant with respect to parabolic scaling. Thus, in the following when we say mixed type Lip(1, α) domains, we mean domains lying above graphs as in (1), which are C 0,1 in space and C 0,α in time, α ∈ (0, 1). Since the works of Kemper, most notably [15], the study of time dependent domains has garnered a lot of attention.…”

“…We assume that u vanishes continuously on a part of the boundary and we study the type of geometric assumptions we need to pose on the boundary in order to prove the Carleson estimate for (2), initially proved for equations of p-parabolic type by the author together with Ugo Gianazza and Sandro Salsa for cylindrical Lipschitz domains, in [1]. In the linear ( p = 2) case, the Carleson seems to be very useful, see for example [9][10][11][12][13][14][15]18,20], thus it seems meaningful to establish it for p > 2.…”

“…In Sects. 5, and 6 we use a modified version of the proof in [1] to prove the Carleson estimate for intrinsic parabolic NTA domains.…”