Boundary estimates for solutions to linear degenerate parabolic equations

Abstract

Let Ω ⊂ R n be a bounded NTA-domain and let Ω T = Ω × (0, T ) for some T > 0. We study the boundary behaviour of non-negative solutions to the equationWe assume that A(x, t) = {a ij (x, t)} is measurable, real, symmetric and thatfor some constant β ≥ 1 and for some non-negative and real-valued function λ = λ(x) belonging to the Muckenhoupt class A 1+2/n (R n ). Our main results include the doubling property of the associated parabolic measure and the Hölder continuity up to the boundary of quotients of non-negative solutions which vanish continuously on a portion of the boundary. Our results generalize previous results of Fabes, Kenig, Jerison, Serapioni, see [18], [19], [20], to a parabolic setting.

Weighted-W^{1,p} estimates for weak solutions of degenerate and singular elliptic equations

“…In addition, Harnack's inequality and Hölder's regularity of weak solutions were obtained in [14] by adapting the Möser's iteration technique to the non-uniformly elliptic equation (1.1). Since then, Hölder's regularity theory of weak solutions for linear, nonlinear degenerate elliptic and parabolic equations have been extensively developed in [15,19,28,29,38,40] by using and extending ideas and techniques in [14]. See also the earlier paper [41] on Gehring-type gradient estimate for solution of degenerate elliptic equations Sobolev type regularity theory for weak solutions of (1.1) have also been the focus of studied in the past but mostly for the uniformly elliptic case, i.e.…”

Weighted Calderón-Zygmund Estimates for Weak Solutions of Quasi-Linear Degenerate Elliptic Equations

“…This work also extends the recent work [8] to the nonlinear case and two weighted estimates. Results of this paper can be considered as the Sobolev's regularity counterpart of the Schauder's one established in [15,16,22,30,31,35,38,41,42] for singular, degenerate equations.…”

“…See, for instance [1,3,5,6,7,8,9,12,13,23,24,25,26,27,28,29,21,44] for other work in the same directions but only for linear equations or for equations in which A is independent on u. On the other hand, this work includes the case that A could be singular or degenerate as a weight in some Munkenhoupt class of weights as considered in many papers such as [15,16,22,30,31,35,38,41,42] in which only Schauder's regularity of weak solutions are investigated. Moreover, even for the uniformly elliptic case, the results in this paper also improve those in [2,17,36,37] since they do not require a-priori boundedness of weak solutions of (1.1).…”

Quantitative a priori estimates for fast diffusion equations with Caffarelli–Kohn–Nirenberg weights. Harnack inequalities and Hölder continuity

“…We will consider a fixed cylinder of reference Q = (T 0 , T ] × Ω ⊆ (0, ∞) × R N , as in Definition 1.1, and the estimates will take place on a smaller cylinder, typically sufficiently far from the boundary of Ω. Due to the lack of translation invariance of the weights, we need to find the right quantity that takes into account for the change of geometry, following [26,77,87,88] we define for any x 0 ∈ R N and any R > 0:…”