The tusk condition and Petrovskiĭ criterion for the normalized p‐parabolic equation

Abstract

We study boundary regularity for the normalized p‐parabolic equation in arbitrary bounded domains. Effros and Kazdan (Indiana Univ. Math. J. 20 (1970) 683–693) showed that the so‐called tusk condition guarantees regularity for the heat equation. We generalize this result to the normalized p‐parabolic equation, and also obtain Hölder continuity. The tusk condition is a parabolic version of the exterior cone condition. We also obtain a sharp Petrovskiĭ criterion for the regularity of the latest moment of a domain. This criterion implies that the regularity of a boundary point is affected if one side of the equation is multiplied by a constant.

Wiener's criterion for degenerate parabolic equations

“…Proof. If Ω is bounded, we can prove (i) by adapting the proof of Theorem 2.10 in [8] by using Lemma 4.4. In addition, Lemma 4.4 can be deduced by the condition (4.2) holds if u and v are bounded, so (i) also holds under the same condition.…”

“…By using the parabolic modification, it is quite classical to show that H f and H f are solutions to L a (ref the proof of Theorem 5.1 in [8]).…”

Boundary Regularity for the Porous Medium Equation

“…In 1924, Lebesgue [34] characterized regular boundary points for harmonic functions by the existence of barriers. The corresponding characterization for the heat equation was given by Bauer [8] in 1962, but barriers had then already been used to study boundary regularity for the heat equation since Petrovskiȋ [39] in 1935; see the introduction in [11] for more on the history of boundary regularity for the heat equation.…”