Pointwise estimates for the heat equation. Application to the free boundary of the obstacle problem with Dini coefficients

Abstract

We study the pointwise regularity of solutions to parabolic equations. As a first result, we prove that if the modulus of mean oscillation of ∆u − u t at the origin is Dini (in L p average), then the origin is a Lebesgue point of continuity (still in L p average) for D 2 u and ∂ t u. We extend this pointwise regularity result to the parabolic obstacle problem with Dini right hand side. In particular, we prove that the solution to the obstacle problem has, at regular points of the free boundary, a Taylor expansion up to order two in space and one in time (in the L p average). Moreover, we get a quantitative estimate of the error in this Taylor expansion. Our method is based on decay estimates obtained by contradiction, using blow-up arguments and Liouville type theorems. As a by-product of our approach, we deduce that the regular points of the free boundary are locally contained in a C 1 hypersurface for the parabolic distance x 2 + |t|.

Optimal regularity for the obstacle problem for the p-Laplacian

“…In the papers [20] and [21], the right-hand side is allowed to be merely in L p . In [28], the elliptic part of the operator is allowed to be fully nonlinear.…”

Pointwise estimates for degenerate Kolmogorov equations with $L^p$-source term

“…The aim of this paper is to study pointwise regularity of solutions to problem (1.1) for Kolmogorov equations with right hand side in L p . This work may be seen as a generalisation of [13] and [9] where this kind of results are obtained for elliptic and parabolic equations respectively. However, up to our knowledge, the case of Kolmogorov type operators has not been investigated.…”

Higher order parabolic boundary Harnack inequality in <i>C</i>¹ and <i>C</i>^{<i>k</i>, <i>α</i>} domains

“…Without loss of generality assume that ν = e n . Having the bound ||u|| C 1,1 p (Q1) ≤ C, together with [16,Theorem 1.7] and Arzela-Ascoli theorem imply that v converges to its blow up at any free boundary point (x 0 , t 0 ) ∈ ∂Ω ∩ Q r uniformly (independently of the free boundary point) in C 0,1 p , namely 1…”

“…Choose a point (x, t) ∈ Ω ∩ Q r , such that (x, τ ) ∈ Ω c for τ ≤ s. Let (x, s), (x , F (x , t), t) be free boundary points. Therefore by [16,Theorem 1.6] we have v(x, t) ≤ C|x n − F (x , t)| 2 . But since F ∈ C 1 p , and…”