Pointwise Estimates for Laplace Equation. Applications to the Free Boundary of the Obstacle Problem with Dini Coefficients

Abstract

In this paper we are interested in pointwise regularity of solutions to elliptic equations. In a first result, we prove that if the modulus of mean oscillation of ∆u at the origin is Dini (in L p average), then the origin is somehow a Lebesgue point of continuity (still in L p average) for the second derivatives D 2 u. We extend this pointwise regularity result to the obtacle problem for the Laplace equation with Dini right hand side at the origin. Under these assumptions, we prove that the solution to the obstacle problem has a kind of Taylor expansion up to the order 2 (in the L p average). Moreover we get a quatitative estimate of the error in this Taylor expansion for regular points of the free boundary. In the case where the right hand side is moreover double Dini at the origin, we also get a quatitative estimate of the error for singular points of the free boundary. Our method of proof is based on some decay estimates obtained by contradiction, using blow-up arguments and Liouville Theorems. In the case of singular points, our method uses moreover a refined monotonicity formula.

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

“…Following the proof of Lemma 2.9 in [13], we consider a dyadic decomposition of the cylinder Q − ρ , and estimate the quantities in each sub-cylinder. More precisely, we get for 1 ≤ ρ = 2 k r with r ∈ [1/2, 1) that…”

“…Given (u, f ) and λ ∈ (0, 1), let us introduce the notation Contrarily to what is done in [13], the functions N and ω are not necessarily monotone in r. Nevertheless, we have the following routine result (the analogue to Lemma 3.4 in [13]). for some constants C 0 > 0, λ, µ ∈ (0, 1) and assume that ω is Dini.…”

“…Then there exist α ∈ (0, 1] and constants r * ∈ (0, 1], C > 0, such that the following holds. If u ∈ L p (Q − 1 ) satisfies (1.1) with the associatedω defined in (1.2), then we have: i) Pointwise BMO estimate iii) Pointwise control on the solution Ifω is Dini, thenÑ(u, ·) is Dini, and there exists a caloric polynomial P 0 (i.e., a solution of (P 0 ) t = ∆P 0 ) of degree less than or equal to two in space and of degree less than or equal to one in time, such that for every r ∈ (0, r * ] there holds Remark 1.5 Notice that our definition ofω(r) differs from the analogue given in [13], not only because we consider here the parabolic problem instead of the elliptic one, but also because there is no supremum in this new definition. From that point of view, estimate (1.5) is finer than the one given in [13], and than the ones that can be found in the classical literature.…”

A two-phase free boundary problem for harmonic measure

“…Monneau, [Mon09], observed that if f is a harmonic function vanishing to first order at x 0 and p is a 1-homogenous polynomial then M x 0 is monotonically decreasing as r ↓ 0.…”

The classical obstacle problem with coefficients in fractional Sobolev spaces

“…In the same paper (under the same hypotheses of coefficients) the three authors proved a generalization of the monotonicity formula introduced by Monneau [36] to analyze the behaviour near the singular points (see Definition 6.6). In [37] he improved his result; he showed that his monotonicity formula holds under the hypotheses that A ≡ I n and f with a Dini modulus of continuity in an L p sense. In [24] Garofalo and Petrosyan showed the formula of Monneau for the thin obstacle with a regular obstacle.…”