On the Hölder regularity of signed solutions to a doubly nonlinear equation. Part II

Abstract

We demonstrate two proofs for the local Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype isThe first proof takes advantage of the expansion of positivity for the degenerate, parabolic p-Laplacian, thus simplifying the argument; whereas the other proof relies solely on the energy estimates for the doubly nonlinear parabolic equations. After proper adaptions of the interior arguments, we also obtain the boundary regularity for initial-boundary value problems of Dirichlet type and Neumann type.

Local Hölder regularity for nonlocal parabolic $p$-Laplace equations

“…N.Liao in [30] has effected a direct proof of Hölder regularity for parabolic p-Laplace equation by using the expansion of positivity. It has also been used to give the first proofs of Hölder regularity for signchanging solutions of doubly nonlinear equations of porous medium type in [4,9,31].…”

On the Hölder regularity of signed solutions to a doubly nonlinear equation. Part III

“…Here E T := E × (0, T ] for some T > 0 and some E open in R N . Indeed, we have studied the borderline case, i.e., p > 1 and q = p − 1 in [1]; whereas the doubly degenerate case p > 2 and 0 < q < p − 1 has been treated in [2]. In this note, we will push the front line forward by taking on the doubly singular case 1 < p < 2 and 0 < p − 1 < q.…”

“…Therefore, a challenge here is the delicate, technical nature of the intrinsic scaling argument. Furthermore, it is worth pointing out that, like in our previous works [1,2], we dispense with any kind of logarithmic type energy estimate, cf. [5, p. 28].…”

“…The following energy estimate for local weak sub(super)-solutions defined in Section 1.2 has been presented in [2], which actually holds for all p > 1 and q > 0.…”