An extensive study of the regularity of solutions to doubly singular equations

Abstract

In recent years, many papers have been devoted to the regularity of doubly nonlinear singular evolution equations. Many of the proofs are unnecessarily complicated, rely on superfluous assumptions or follow an inappropriate approximation procedure. This makes the theory unclear and quite chaotic to a nonspecialist. The aim of this paper is to fix all the misprints, to follow correct procedures, to exhibit, possibly, the shortest and most elegant proofs and to give a complete and self-contained overview of the theory.

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

“…2) in the case 1 < p < 2 and 0 < p − 1 < q (see Figure 1.1), has been considered by a number of authors, cf. [4,8,9,12]. However, the theory is fragmented in the sense that either they further restrict the range of p and q, or they consider non-negative solutions only.…”

A new short proof of regularity for local weak solutions for a certain class of singular parabolic equations

“…The method we present here can also be implemented for solutions to doubly nonlinear equations of the kind (0.2). The expansion of positivity was proved in [11] (see also [10] and [25]). Equations as (0.2) are the natural bridge between the porous media equations and p-Laplace type ones.…”

“…These equations have been introduced by J.L.Lions in [19] and they describe several physical phenomena; see the survey of A.S.Kalashnikov [16] for more details, see also the following papers for a non-comprehensive surveys on this argument, [17], [18], [21] and [22]. In this paper we take as a starting point the recent extensive study made in [25] and we also refer to it for a self-contained introduction to the regularity theory for doubly nonlinear equations.…”

“…Then there is a time level t where in the ball B we have |u(•, t) > 1 2 ] ∩ B| > ν|B|. Let us apply an integral Harnack estimate introduced for the first time for the p-Laplacean in [1] (see also [8]) and for the doubly nonlinear case in [10] (see also [25]). Thanks to this inequality, the measure information can be extended to any time level in Q.…”

Harnack’s inequality for doubly nonlinear equations of slow diffusion type

“…A weak Harnack inequality for super-solutions can be found in [24]. For the case of fast diffusion equations, we refer to the articles by Fornaro, Sosio and Vespri [13] and Vespri and Vestberg [34].…”