Boundary regularity for degenerate and singular parabolic equations

Abstract

We characterise regular boundary points of the parabolic $p$-Laplacian in terms of a family of barriers, both when $p>2$ and $1<p<2$. Due to the fact that $p\not=2$, it turns out that one can multiply the $p$-Laplace operator by a positive constant, without affecting the regularity of a boundary point. By constructing suitable families of barriers, we give some simple geometric conditions that ensure the regularity of boundary points

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

“…that is if u is a solution to (2) then v is also a solution to (2), thus it is reasonable to expect that the regularity of the boundary is determined to some extent by the solution itself. Usually one combines the two scalings to form one scaling,…”

“…to obtain a form for which most estimates for solutions to (2) have statements which are homogeneous and scale invariant. At this point in time not much is known about boundary behavior of solutions to (2) in time dependent domains, basically all that is known can be found in the following papers [2,16,19].…”

“…At this point in time not much is known about boundary behavior of solutions to (2) in time dependent domains, basically all that is known can be found in the following papers [2,16,19].…”

Regularity of \( p\left(\right.\cdot\left.\right) \)-superharmonic functions, the Kellogg property and semiregular boundary points

“…In fact, the situation is similar for p-parabolic equations in the sense that if u is a p-parabolic function and a ∈ R, then u + a is p-parabolic, but au is in general not p-parabolic. In the p-parabolic case a similar characterization of boundary regularity to the one above was obtained by Björn, Björn, Gianazza, and Parviainen [10]. Therein a characterization of boundary regularity in terms of the existence of a family of barriers was also obtained.…”

A comparison principle for the porous medium equation and its consequences

“…Such a result is occasionally called the elliptic comparison principle, in reference to the fact that the time variable no longer has a special role. Moreover, the elliptic comparison principle can be used to develop the Perron method in general space-time domains, see [3,12]. We also present two applications where such a comparison principle is indispensable.…”