E Lindgren
email: erik.lindgren@math.ntnu.no

R Monneau

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

Keywords: AMS Classification: 35R35 Obstacle problem, Heat equation, Dini condition, free boundary, pointwise regularity

We study the pointwise regularity of solutions to parabolic equations. As a first result, we prove that if the modulus of mean oscillation of ∆uu 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|.

1 Introduction

The heat equation

In this paper, we are interested in the pointwise regularity of solutions to parabolic problems. We first consider the solutions to the following heat equation

(1.1)    ∆u -u t = f in Q - 1 , f ∈ L p Q - 1 and f (0) = 0,
where we set the past cylinder Q - r = B r × (-r 2 , 0] with B r = B r (0) the open ball in R n , of radius r centered at the origin 0. Here p ∈ (1, +∞) and we assume that 0 = (0, 0) is a Lebesgue point of f , in order to define f (0) if necessary.

It is well-known that if f is Hölder continuous in the cylinder Q - 1 , then so are the spatial second derivatives of u and the time first derivative of u (see for instance [START_REF] Gary | Second order parabolic differential equations[END_REF]). Let us introduce the following parabolic modulus of continuity of f on the cylinder Q - It is well-known (see [START_REF] Wang | Schauder Estimates for Elliptic and Parabolic equations[END_REF]) that if σ is Dini, then the second derivatives of u are continuous in the cylinder Q - 1/2 with a modulus of continuity proportional to r sup

Q - 1 |u| + r 0 σ(s) s ds + r 1 r σ(s) s 2 ds.
Notice that the modulus of continuity of u t then follows from equation (1.1) itself.

Up to our knowledge, such results are usually obtained assuming a modulus of continuity in an open set. Here we change the point of view, and only consider pointwise modulus of mean oscillation, like for instance [START_REF] Zou | Fully Nonlinear Parabolic Equations and the Dini Condition[END_REF]. For any p ∈ (1, +∞), we define a kind of modulus of mean oscillation (in L p average) of the function f at the origin as

(1.2) ω(r) = ω(f, r) = inf c∈R 1 |Q - r | Q - r |f (x, t) -c| p 1 p
.

Furthermore, we denote by P2 the set of polynomials of degree less than or equal to two in space and of degree less than or equal to one in time. Let

(1.3) Ñ(u, ρ) = inf P ∈ P2 1 r n+2+2p Q - r |u -P | p 1 p
. Theorem 1.2 (Pointwise parabolic BMO estimates for the heat equation) Let p ∈ (1, +∞). 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 associated ω defined in (1.2), then we have: i) Pointwise BMO estimate (1.4) sup iii) Pointwise control on the solution If ω is Dini, then Ñ(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

r∈(0,1] Ñ(u, r) ≤ C    Q - 1 |u| p 1 p + Q - 1 |f | p
(1.5) 1 |Q - r | Q -
s ds + Q - 1 |u| p 1 p + Q - 1 |f | p 1 p
.

Remark 1.3 Theorem 1.2 iii) implies in particular (using parabolic estimates) that we have a Lebesgue point of continuity of the second derivatives D 2 u and of u t (in the L p average) if ∆u -u t has a Dini modulus of mean oscillation (in the L p average) at the same point.

Remark 1.4 A straightforward consequence of Theorem 1.2 is that the second derivatives D 2 u and u t are Hölder continuous in an open set Ω ⊂ R n+1 , if ∆u -u t is Hölder continuous in Ω for the parabolic distance |x| 2 + |t|.

Remark 1.5 Notice that our definition of ω(r) differs from the analogue given in [START_REF] Monneau | Pointwise estimates for Laplace equation. Applications to the free boundary of the obstacle problem with Dini coefficients[END_REF], 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 [START_REF] Monneau | Pointwise estimates for Laplace equation. Applications to the free boundary of the obstacle problem with Dini coefficients[END_REF], and than the ones that can be found in the classical literature.

We would like to emphasize that the result of Theorem 1.2 is completely pointwise, which does not seem to be so usual in the literature.

The model obstacle problem

In the second part of this article we are in particular interested in the regularity of the free boundary for solutions to the parabolic obstacle problem. The model problem is the following. Consider a function u satisfying

(1.6)                    ∆u -u t = f (x, t)χ {u>0} u ≥ 0 in Q - 1 , u, f ∈ L p (Q - 1 ) and f (0) = f (0, 0) = 1, 0 ∈ ∂ {u > 0} ,
for p ∈ ((n+2)/2, +∞), where Q - 1 is the past unit cylinder as before and χ {u>0} is the characteristic function of the set {u > 0}, which is equal to 1 if u > 0 and 0 if u = 0. From classical parabolic estimates joint with Sobolev embeddings with our assumption p > (n + 2)/2, every solution u is in particular continuous, which allows us to consider the boundary of the open set {u > 0}. Here ∂ {u > 0} is called the free boundary. Moreover, we assume that (0, 0) is a Lebesgue point for f in order to define f (0).

There is a vast literature on the above problem. In the special case when f = 1 and in a slightly more general setting, it is proved in [START_REF] Apushkinskaya | Boundary estimates for solutions of a parabolic free boundary problem[END_REF], that the solution enjoys the optimal C 1,1

x ∩ C 0,1 t regularity. Moreover, in [START_REF] Apushkinskaya | On the Lipschitz property of the free boundary in a parabolic problem with an obstacle[END_REF], it is proved that the free boundary ∂{u > 0} is, close to the part of the fixed boundary where u satisfies a homogeneous Dirichlet condition, the graph of a Lipschitz function. This was extended to a more general problem in [START_REF] Caffarelli | Regularity of a free boundary in parabolic potential theory[END_REF], where it is proved that the free boundary is at regular points a C 1 regular graph. Some partial regularity results are also proved in [START_REF] Edquist | Regularity of a parabolic free boundary problem with hölder continuous coefficients[END_REF], under the assumptions that f is Hölder continuous.

In the one dimensional setting, and under the assumption that f is Dini continuous, there is a series of paper, [START_REF] Blanchet | On the one-dimensional parabolic obstacle problem with variable coefficients, Elliptic and parabolic problems[END_REF], [START_REF] Blanchet | On the continuity of the time derivative of the solution to the parabolic obstacle problem with variable coefficients[END_REF] and [START_REF] Blanchet | On the singular set of the parabolic obstacle problem[END_REF], where this problem is studied. There it is proved that the free boundary is C 1 regular at certain regular (see the next page) points, and also that the free boundary enjoys a certain structure at the other points, the so called singular points.

Let us introduce the following kind of pointwise modulus of continuity (in L p average) of the function f at the origin:

(1.7) σ(r) = σ(f, r) = sup 0<ρ≤r ω(ρ) with ω(ρ) = ω(f, ρ) = 1 |Q - ρ | Q - ρ |f (x, t) -f (0)| p 1 p
We have the following general regularity result.

Proposition 1.6 (Quadratic growth) Let p ∈ ((n + 2)/2, +∞). Then there exists a constant C > 0 such that if u is a solution of (1.6) with σ bounded given by (1.7), then

0 ≤ u(x, t) ≤ C 1 |x| 2 + |t| in Q - 1 2
,

where

C 1 = C (1 + σ(1)).
In order to present our main result, we need to introduce the quantity

M reg (u, r) = sup ρ∈(0,r]   inf P ∈Preg 1 ρ n+2+2p Q - ρ |u -P | p 1 p   ,
where

P reg = P (x, t) = 1 2 (max(0, x • ν)) 2 , ν ∈ S n-1 .
Notice that as a consequence of Proposition 1.6, M reg (u, ρ) is bounded for ρ ≤ 1/2. Recall that if the free boundary is smooth (or regular) around the origin, then it is known that the blow-up limit of the solution (i.e., the limit of certain rescalings of the solution) at the origin is unique and is an element of the set P reg . Therefore we have in particular

(1.8) lim r→0 + M reg (u, r) = 0.
More generally, we define the set of regular points as

R =        (x 0 , t 0 ) ∈ Q - 1 , (x 0 , t 0 ) is a Lebesgue point of f with f (x 0 , t 0 ) > 0 and lim r→0 + M reg 1 f (x 0 , t 0 ) u(x 0 + •, t 0 + •), r = 0       
.

Our main result is the following:

Theorem 1.7 (Modulus of continuity at a regular point of the free boundary) Let p ∈ ((n + 2)/2, ∞). There exist α ∈ (0, 1] and constants C > 0,M 0 , r 0 ∈ (0, 1) such that, given u satisfying (1.6), we have the following property. If the modulus of continuity σ defined in (1.7) is assumed Dini, and if

M reg (u, r 0 ) ≤ M 0 ,
then there exists P 0 ∈ P reg such that for every r ∈ (0, r 0 )

1 |Q - r | Q - r u -P 0 r 2 p 1 p ≤ C M reg (u, r 0 )r α + r 0 σ(s) s ds + r α 1 r σ(s) s 1+α ds .
Remark 1.8 With the same methods, it would be possible to get a similar estimate for any p ∈ (1, +∞), but under the stronger assumption that the coefficient of the right hand side of the equation is bounded from above and from below, i.e., 0 < δ 0 ≤ f ≤ 1/δ 0 .

Remark that under the assumptions of Theorem 1.7, we recover in particular (1.8). As a corollary of Theorem 1.7 and using a Weiss type monotonicity formula, we will show in a companion paper [START_REF] Lindgren | Pointwise estimates at singular points for the parabolic obstacle problem[END_REF] the result below.

Theorem 1.9 (Regularity of the regular set of the free boundary, [START_REF] Lindgren | Pointwise estimates at singular points for the parabolic obstacle problem[END_REF]) Consider a solution u of (1.6), and assume that σ defined in (1.8) is Dini with f ≥ δ 0 > 0 on Q - 1 . Then for any point (x 0 , t 0 ) ∈ R, there exists a neighborhood V of (x 0 , t 0 ) in Q - 1 , such that V ∩ ∂ {u > 0} is locally a C 1 hypersurface with respect to the parabolic distance. More precisely, up to a rotation of the spatial coordinates

V ∩ ∂ {u > 0} = {(x, t) such that x n = g(x ′ , t) with (x ′ , t) ∈ V ′ } , where x ′ = (x 1 , ..., x n-1 ), the set V ′ is an open set in R n , and g : V ′ → R is a map satisfying g(x ′ + h ′ , t + k) = g(x, t) + h ′ • D x ′ g(x ′ , t) + o( (h ′ ) 2 + |k|), with D x ′ g continuous on V ′ .
In [START_REF] Lindgren | Pointwise estimates at singular points for the parabolic obstacle problem[END_REF] we will also present a theory for the singular points of the free boundary, that is, for the complement of the regular part.

Organization of the paper

The organization of the paper is as follows. First, in Section 2, we recall certain classical results concerning parabolic Sobolev spaces and parabolic equations. This is followed by Section 3, where we, by contradictory and blow-up type arguments, prove our main result for the heat equation, namely Theorem 1.2.

In Section 4 we turn our attention to the obstacle problem. We prove, using mainly standard techniques, quadratic growth estimates for the obstacle problem and in Section 5, we exploit a standard non-degeneracy result and obtain a related, somewhat more technical result, refered to as weak non-degeneracy. In the following section, namely Section 6, we provide a compactness result that we strongly use to prove the main theorem (Theorem 1.7) for the obstacle problem, which is proved, using contradictory and blow-up type arguments, in Section 7.

Notation

Throughout the whole paper we will use the notation below:

u t = ∂ t u = ∂u ∂t -the time derivative ∆u = n i=1 ∂ 2 u ∂x i 2 -the Laplace operator Hu := ∆u -u t -the heat operator Q r (x 0 , t 0 ) = B r (x 0 ) × (t 0 -r 2 , t 0 + r 2 ) -a parabolic cylinder Q - r (x 0 , t 0 ) = B r (x 0 ) × (t 0 -r 2 , 0] -a half cylinder ∂ p Q - r (x 0 , t 0 ) = ((∂B r (x 0 )) × [t 0 -r 2 , 0)) (B r (x 0 ) × {0}) -the parabolic boundary Q r = Q r (0, 0), Q - r = Q - r (0, 0), ∂ p Q - r = ∂ p Q - r (0, 0) -simplified notation ω(g, ρ) = 1 |Q - ρ | Q - ρ |g(x, t) -g(0)| p 1 p
-the average oscillation over a cylinder σ(g, r) = sup 0<ρ≤r ω(g, ρ)

-a special L p -modulus P2 -polynomials of parabolic degree less than or equal to two P 2 -P ∈ P2 such that HP = 0 P 2,c -P ∈ P2 such that HP = c 2 Classical results for parabolic equations

Here we recall the following classical results that will be of constant use in the rest of the paper.

Theorem 2.1 (Parabolic interior L p -estimates) Let p ∈ (1, ∞). If u ∈ L p (Q - r ) and Hu ∈ L p (Q - r ) then ||u|| W 2,1 p (Q - r/2 ) ≤ C ||u|| L p (Q - r ) + ||Hu|| L p (Q - r ) , where W 2,1 p (Q - ρ ) = {v ∈ L p (Q - ρ ) v, ∇v, D 2 v, v t ∈ L p (Q - ρ )}, endowed with the norm ||u|| W 2,1 p (Q - ρ ) = ||u|| L p (Q - ρ ) + ||∇u|| L p (Q - ρ ) + ||D 2 u|| L p (Q - ρ ) + ||u t || L p (Q - ρ ) .
The result above is a special case of Theorem 7.22 on page 175 in [START_REF] Gary | Second order parabolic differential equations[END_REF].

Theorem 2.2 (Parabolic Sobolev embedding) Let u ∈ W 2,1 p (Q - r ) with p ∈ ((n + 2)/2, ∞). Then ||u|| C α (Q - r ) ≤ C * ||u|| W 2,1 p (Q - r ) ,
with α = 2 -n+2 p and where C α (Q - r ) refers to the parabolic Hölder space.

This result is contained in Lemma 3.3 on page 80 in [START_REF] Ladyženskaja | Linear and quasilinear equations of parabolic type[END_REF].

Theorem 2.3 (Classical L p parabolic estimate) Let u ∈ W 2,1 p (Q - r ) for p ∈ (1, +∞) a solution of Hu = f on Q - r , u = 0 on ∂ p Q - r ,
where

∂ p Q - r denotes the parabolic boundary of Q - r , and f ∈ L p (Q - r ).
Then there exists a constant C > 0 (depending only on p, the dimension n and r > 0) such that

||u|| W 2,1 p (Q - r ) ≤ C||f || L p (Q - r ) .
This result can be found in Proposition 7.18 on page 173 in [START_REF] Gary | Second order parabolic differential equations[END_REF].

3 Proof of Theorem 1.2

In order to give the proof of Theorem 1.2, we show a basic decay estimate in a first subsection and some routine results in a second subsection. The proof of Theorem 1.2 is done in the third subsection.

A basic decay estimate

Given a function f , we consider a (unique

) constant c r such that ω(f, r) = inf c∈R 1 |Q - r | Q - r |f (x, t) -c| p 1 p = 1 |Q - r | Q - r |f (x, t) -c r | p 1 p
.

We define the particular set of caloric polynomials:

P 2 =    P a caloric polynomial
of degree less than or equal to 2 in space of degree less than or equal to 1 in time

  
Considering a particular polynomial P * ∈ P2 which satisfies ∆P * -(P * ) t = 1 (for instance

P * (x, t) = x 2 2n
), we define P 2,c = cP * + P 2 and for a function u solving (1.1) we let

(3.1) N(u, r) = inf P ∈P 2,cr 1 r n+2+2p Q - r |u -P | p 1 p .
For ω(f, r) and N(u, r) respectively defined in (1.2) and (3.1), we now define for 0 < a < b

(3.2) N (u, a, b) = sup a≤ρ≤b N(u, ρ) and ω(f, a, b) = sup a≤ρ≤b ω(f, ρ).
Then we have the decay estimate below.

Proposition 3.1 (Basic decay estimate)

Given p ∈ (1, +∞), there exist constants C 0 > 0 λ, µ ∈ (0, 1) (depending on p and the dimension n) such that for every function u and f satisfying (1.1) with the notation given in (3.2), there holds

(3.3) ∀r ∈ (0, 1], N (u, λ 2 r, λr) < µ N (u, λr, r) or N (u, λ 2 r, λr) < C 0 ω(f, λ 2 r, r).
In order to prove this proposition, we will need the following result whose proof is postponed to subsection 3.2.

Lemma 3.2 (Estimates of N in larger balls) Let u be a solution of ∆u -u t = f in B R for R > 2, and N(u, 1) = ||u -P 1 || L p (Q - 1 )
, with P 1 ∈ P 2,1 . Then for any ρ ∈ [1, R/2], we have

(3.4) 1 ρ n+2+2p Q - ρ |u -P 1 | p dx dt 1 p ≤ C 1 2ρ 1 N(u, s) + ω(f, s) s ds.

Proof of Proposition 3.1

The proof is done by contradiction. If this is not true, we can find sequences

C k → ∞, r k ∈ (0, 1], λ k → 0 and µ k → 1 such that (3.
3) fails with the corresponding functions u k and f k satisfying (1.1). This means that

(3.5)    N(u k , λ 2 k r k , λ k r k ) ≥ µ k N (u k , λ k r k , r k ), N(u k , λ 2 k r k , λ k r k ) ≥ C k ω(f k , λ 2 k r k , r k ).
Step 1: Construction of sequences and a priori estimates Let us consider a (not necessarily unique)

ρ k ∈ [λ 2 k r k , λ k r k ] so that N(u k , λ 2 k r k , λ k r k ) = N (u k , ρ k ) =: ε k .
Moreover, define the rescaled functions

v k (x, t) = u k (ρ k x, ρ 2 k t) ρ 2 k and (3.6) w k (x, t) = u k (ρ k x, ρ 2 k t) -P k (ρ k x, ρ 2 k t) ε k ρ 2 k ,
where P k ∈ P 2,cρ k is one polynomial realizing the infimum defining N(u k , •) at the level ρ k . Now, we wish to pass to the limit, but first we need to control the sequence w k . By definition (3.7) inf

P ∈P 2 Q - 1 |w k -P | p dx dt 1 p = 1.
In addition, since

N(u k , 1) = 1 ρ n+2+2p k Q - ρ k |u k -P k | p dx dt 1 p , we also have for s ∈ (1, r k 2ρ k ) (applying Lemma 3.2 on v k ) 1 s n+2+2p Q - s |w k | p dx dt 1 p = 1 ε k 1 (sρ k ) n+2+2p Q - sρ k |u k -P k | p dx dt 1 p ≤ C 1 ε k 2s 1 N(u k , τ ρ k ) + ω(f k , τ ρ k ) τ dτ ≤ C 1 ε k 2s 1 N(u k , λ 2 k r k , r k ) + ω(f k , λ 2 k r k , r k ) τ dτ.
Notice that from (3.5) we deduce

N(u k , λ 2 k r k , r k ) = max( N (u k , λ 2 k r k , λ k r k ), N(u k , λ k r k , r k )) ≤ ε k µ k and ω(f k , λ 2 k r k , r k ) ≤ ε k C k . This implies for s ∈ (1, r k 2ρ k ) and some constant C 2 > 0 (3.8) 1 s n+2+2p Q - s |w k | p dx dt 1 p ≤ C 2 ln 2s.
Furthermore, one can easily check that for

H = ∆ -∂ t and s ∈ (1, r k 2ρ k ) we have 1 |Q - s | Q - s |Hw k | p 1 p ≤ 1 ε k ω(f k , sρ k ) ≤ 1 ε k ω(f k , λ 2 k r k , r k ) ≤ 1 C k → 0.
Step 2: Identifying the limit and contradiction From (3.8) and the interior parabolic estimate (Theorem 2.1), it follows that there is a subsequence again labeled w k , converging in L p loc (R n × R -) to a caloric function w 0 . By passing to the limit in (3.7) we get (3.9) inf

P ∈P 2 Q - 1 |w 0 -P | p dx dt 1 p = 1.
Similarly, passing to the limit in (3.8) yields for all s ≥ 1

1 s n+2+2p Q - s |w 0 | p dx dt 1 p ≤ C 2 ln 2s.
Hence, w 0 is a caloric function in R n × R -that grows at most quadratically in space and linearly in time (up a logarithmic correction). This implies that w 0 is a caloric polynomial of degree at most two in space and one in time, i.e. w 0 ∈ P2 . This clearly contradicts (3.9). This ends the proof of Proposition 3.1.

Some routine results

Proof of Lemma 3.2

The proof of this lemma is similar to the proof of Lemma 2.9 in [START_REF] Monneau | Pointwise estimates for Laplace equation. Applications to the free boundary of the obstacle problem with Dini coefficients[END_REF].

Step 1: Statement of (3.10) On the one hand, we use the fact that there exists a constant C 2 > 0 such that for any P ∈ P2 and for any r ≥ 1 there holds

1 r n+2+2p Q - r |P | p 1 p ≤ C 2 Q - 1 |P | p 1 p .
Following the proof of Lemma 2.9 in [START_REF] Monneau | Pointwise estimates for Laplace equation. Applications to the free boundary of the obstacle problem with Dini coefficients[END_REF], 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 (3.10) 1 ρ n+2+2p Q - ρ |u -P 1 | p dx dt 1 p ≤ C 3 N (u, 1) + k j=0 N (u, 2 j r) .
Step 2: Proof of estimate (3.11) On the other hand, for any γ > 1, we also notice that for α ∈ [1, γ], we have for any r > 0

N (u, αr) ≤ γ 2+ n+2 p N (u, γr) + |c αr -c γr | Q - 1 |P * | p 1 p , and          |c αr -c γr | = 1 |Q - αr | Q - αr |c αr -c γr | p 1 p ≤ ω(f, αr) + γ n+2 p ω(f, γr), ω(f, αr) ≤ γ n+2 p ω(f, γr).
Therefore, for any γ > 1, there exists a constant C γ > 0 such that

(3.11) ∀α ∈ [1, γ],      N (u, αr) ≤ C γ N (u, γr) + ω(f, γr) , ω(f, αr) ≤ C γ ω(f, γr).
Step 3: Conclusion Using (3.11) with γ = 2, we get the result (3.4) with the integral on N replacing the sum in the right hand side of (3.10). This ends the proof of Lemma 3.2.

Given (u, f ) and λ ∈ (0, 1), let us introduce the notation (3.12)

N (r) = N(u, λr, r) and ω(r) = ω(f, λ 2 r, r).
Contrarily to what is done in [START_REF] Monneau | Pointwise estimates for Laplace equation. Applications to the free boundary of the obstacle problem with Dini coefficients[END_REF], 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 [START_REF] Monneau | Pointwise estimates for Laplace equation. Applications to the free boundary of the obstacle problem with Dini coefficients[END_REF]).

Proposition 3.3 (Dini estimate)

Let N : (0, 1] → [0, +∞), ω : (0, 1] → [0, +∞) be two functions satisfying

(3.13) ∀r ∈ (0, 1], N(λr) ≤ µN(r) or N (λr) ≤ C 0 ω(r)
and

(3.14) ∀r ∈ (0, 1], ∀α ∈ [λ, 1],    N(αr) ≤ C 0 (N (r) + ω(r)) , ω(αr) ≤ C 0 ω(r),
for some constants C 0 > 0, λ, µ ∈ (0, 1) and assume that ω is Dini. Then there exists a constant C ′ 0 > 0 depending only on C 0 , λ, µ > 0, such that for every ρ ∈ (0, λ/2] and with α = ln µ/ ln λ there holds

C 0 = max(C 0 , C 1/λ )).
Proof of Proposition 3.3

Step 1: Estimate on N(r)

We claim that we have for all r ∈ (0, λ]

(3.15) N(r) ≤ max C 2 r α , C 0 r α sup ρ∈[r,λ] ω(ρ) ρ α .
The proof is the same as Lemma 3.3 in [START_REF] Monneau | Pointwise estimates for Laplace equation. Applications to the free boundary of the obstacle problem with Dini coefficients[END_REF] with r 0 = 1, except that we estimate for r 1 ∈ (λ, 1]

N(r 1 ) ≤ C 0 (N(1) + ω(1)).
This gives the new value to the constant

(3.16) C 2 = λ -α C 0 (N (1) + ω(1)
).

Here we have replaced the lack of monotonicity of N by the first line of assumption (3.14).

Step 2: Estimate on sup ρ∈[r,λ] ω(ρ) ρ α and conclusion We follow the proof of Lemma 3.4 in [START_REF] Monneau | Pointwise estimates for Laplace equation. Applications to the free boundary of the obstacle problem with Dini coefficients[END_REF]. For some ρ 0 ∈ [r, λ] we have

sup ρ∈[r,λ] ω(ρ) ρ α = ω(ρ 0 ) ρ α 0 ≤ 1 ρ α 0 1 tρ 0 ρ 0 +tρ 0 ρ 0 C 0 ω(ρ) dρ with t = 1 -λ λ > 0 ≤ C 0 tλ 1+α ρ 0 λ ρ 0 ω(ρ) ρ 1+α dρ ≤ C 3 1 r ω(ρ) ρ 1+α dρ with C 3 = C 0 (1 -λ)λ α > 0,
where in the second line we have used the second line of assumption (3.14) (because of the lack of monotonicity of ω). The remaining part of the proof of Lemma 3.4 in [START_REF] Monneau | Pointwise estimates for Laplace equation. Applications to the free boundary of the obstacle problem with Dini coefficients[END_REF] is unchanged and then implies the result. This ends the proof of Proposition 3.3.

Proof of

≤ C u L p (Q - 1 ) + f L p (Q - 1 )
.

Proof of ii)

This follows from (3.15).

Proof of iii)

Step 1: Dini property Proposition 3.3 implies that N is Dini if ω is Dini. Then we deduce that N(u, r) (and then Ñ (u, r)) is Dini, if ω(f, r) is Dini.

Step 2: Estimate (1.5)

The proof of Lemma 3.5 in [START_REF] Monneau | Pointwise estimates for Laplace equation. Applications to the free boundary of the obstacle problem with Dini coefficients[END_REF] is straightforward to adapt to our case. Using our Lemma 3.2 instead of Lemma 2.9 in [START_REF] Monneau | Pointwise estimates for Laplace equation. Applications to the free boundary of the obstacle problem with Dini coefficients[END_REF], this shows that there exists a polynomial P 0 ∈ P2 such that for all ρ ∈ (0, r * ] with r * = λ/2

1 ρ n+2+2p Q - ρ |u -P 0 | p dx dt 1 p ≤ C 1 2ρ 1 N(u, s) + ω(f, s) s ds.
We deduce (1.5) using Proposition 3.3 joint with (3.11).

Step 3: P 0 is caloric From (1.5) and the interior estimates (Theorem 2.1) we also deduce that

u ε (x, t) = (u -P 0 )(εx, ε 2 t) ε 2
converges, as ε → 0, to a function v ≡ 0, which is a solution of

Hv = f (0) -H(P 0 ) with H = ∆ -∂ t .
Since f (0) = 0, this shows that P 0 is caloric.

Step 4: Bound on the coefficients of P 0 We simply apply (1.5) for r = λ/2 and this implies the bound on the coefficients of P 0 . This ends the proof of the theorem.

Growth estimates for the obstacle problem

In this section we will prove some growth estimates of solutions to (1.6). Some of the results are of independent interest while some are needed in the sequel.

Proposition 4.1 (Quadratic growth in mean)

Let u be a solution of (1.6) with p ∈ ((n + 2)/2, ∞). Then there are positive constants r 0 and

C 1 = C (1 + σ(1)) u L p (Q - 1 ) , such that 1 |Q - r | Q - r |u| p 1 p ≤ C 1 r 2 ,
whenever r < r 0 .

Proof of Proposition 4.1 Define for r ∈ (0, 1]

S r (u) = 1 |Q - r | Q - r |u| p 1 p
.

By iteration it is sufficient to prove that there exists C > 0 and r 0 ∈ (0, 1] such that for all solutions u of (1.6), for all r ≤ r 0 either

S r (u) ≤ Cr 2 (1 + σ(f, 1)) or S r (u) ≤ 4 -k S 2 k r (u) for some k ∈ N such that 2 k r ≤ 1.
In order to prove that this holds we argue by contradiction. If this does not hold, there are sequences u j , f j and r j → 0,C j → ∞ such that

S r j (u j ) ≥ C j r 2 j (1 + σ(f j , 1)) and S r j (u j ) ≥ 4 -k S 2 k r j (u j ) for all k ∈ N such that 2 k r j ≤ 1.
Define the rescaled functions

v j (x, t) = u j (r j x, r 2 j t) S r j (u j ) . Then 1. Hv j = r 2 j S r j (u j ) f j (r j x, r 2 j t)χ {v j >0} in Q - 1 r j , 2. S 1 (v j ) = 1, 3. S 2 k (v j ) ≤ 4 k for all k ∈ N with 2 k r j ≤ 1, 4. v j (0) = 0, 5. v j ≥ 0.
Observing that for ρ ≤ 1/r j we have

1 |Q - ρ | Q - ρ |f j (r j x, r 2 j t)| p 1 p = 1 |Q - ρr j | Q - ρr j |f j (x, t)| p 1 p ≤ 1 + σ(f j , 1),
we see that Hv j is bounded in L p (Q - ρ ) for every ρ and converges to 0. Therefore, by interior parabolic estimates and the Sobolev embedding (Theorem 2.1 and Theorem 2.2) there exists a subsequence, again labelled v j , such that v j → v 0 locally in C α and locally weakly in W 2,1 p , where v 0 satisfies 1.

Hv 0 = 0 in R n × R -, 2. S 1 (v 0 ) = 1, 3. v 0 (0) = 0, 4. v 0 ≥ 0.
This contradicts the strong maximum principle for caloric functions (see Theorem 11 on page 375 in [START_REF] Evans | Partial differential equations[END_REF]) and ends the proof of the proposition.

Using Proposition 4.1 we prove the following corollary that implies Proposition 1.6.

Corollary 4.2 (Quadratic growth in the sup-norm)

Under the same assumptions as in Proposition 4.1, there is a constant C 2 > 0 such that there holds sup

Q - r |u| ≤ C 2 C 1 r 2 for all r < r 0 /2,
where r 0 and C 1 are defined in Proposition 4.1.

Proof of Corollary 4.2 Define v r (x, t) = u(rx, r 2 t) r 2 .

Then Proposition 4.1 implies that for r < r 0 ,

1 |Q - 1 | Q - 1 |v r | p 1 p ≤ C 1 .
Moreover, there holds

1 |Q - 1 | Q - 1 |Hv r | p 1 p ≤ 1 + σ(1).
Therefore, by interior estimates (Theorem 2.1)

||v r || W 2,1 p (Q - 1 2 ) ≤ C ′ 2 C 1 ,
and thus, by the Sobolev embedding (Theorem 2.2)

sup Q - 1 2 |v r | ≤ C 2 C 1 .
Scaling back to u yields the desired result.

Non-degeneracy

In this section we prove that solutions of (1.6) cannot decay too fast close to the origin, and the rate at which this can happen, naturally depends on f . Proposition 5.1 (Non-degeneracy) Let p ∈ ((n + 2)/2, ∞). In addition, assume that u solves

   Hu = f χ {u>0} u ≥ 0 in Q - R , f (0) = 1.
Then there exists a constant C 0 > 0 such that

sup ∂pQ - d (x 0 ,t 0 ) u ≥ d 2 2n + 1 , whenever Q - d (x 0 , t 0 ) ⊂ Q - R and u(x 0 , t 0 ) > 2λ, where λ = C 0 R n+2 p d 2-n+2 p 1 |Q - R | Q - R |f (x, t) -f (0)| p 1 p
.

Proof of Proposition 5.1

Let v be the solution of

Hv = f -f (0) in Q - d (x 0 , t 0 ), v = 0 on ∂ p Q - d (x 0 , t 0 ). Defining v d (x, t) = v(dx + x 0 , d 2 t + t 0 ) d 2 and f d (x, t) = f (dx + x 0 , d 2 t + t 0 ), we see that Hv d = f d -f (0) in Q - 1 , v d = 0 on ∂ p Q - 1 .
The classical parabolic estimates (Theorem 2.3) imply

||v d || W 2,1 p (Q - 1 ) ≤ C||f d -f (0)|| L p (Q - 1 )
. Applying the Sobolev embedding (Theorem 2.2) and scaling back to v, we deduce sup

Q - d (x 0 ,t 0 ) |v| ≤ d 2 C 0 1 d n+2 |Q - 1 | Q - d (x 0 ,t 0 ) |f (x, t) -f (0)| p 1 p ≤ λ, with C 0 = C * C|Q - 1 | 1 p . Let w = u -v -f (0) (x -x 0 ) 2 -(t -t 0 ) 2n + 1 .
Then Hw = 0 in Q - d (x 0 , t 0 ) ∩ {u > 0}. Moreover, w(x 0 , t 0 ) = u(x 0 , t 0 ) -v(x 0 , t 0 ) > λ. Therefore, by the maximum principle, w attains its positive maximum > λ on ∂ p (Q d (x 0 , t 0 ) ∩ {u > 0}). Whenever u = 0 we have w ≤ λ. Thus, the maximum is attained on {u > 0} ∩ ∂ p Q d (x 0 , t 0 ) and we have λ < sup

{u>0}∩∂pQ d (x 0 ,t 0 ) w ≤ sup {u>0}∩∂pQ d (x 0 ,t 0 ) u - d 2 2n + 1 + λ.

The result follows.

A corollary from this non-degeneracy follows below.

Corollary 5.2 (Weak non-degeneracy)

Let p ∈ ((n + 2)/2, ∞). Assume that u m and f m verify

                     Hu m = f m χ {um>0} u m ≥ 0 in Q - R , f m (0) = 1, τ m = 1 |Q - R | Q - R |f m (x, t) -f m (0)| p 1 p → 0 u m → u ∞ in L ∞ loc (Q - R )
as m → ∞.

Then for any compact

K ⊂ {u ∞ = 0} • ∩ Q - R (where {u ∞ = 0}
• denotes the interior of the set {u ∞ = 0}), there exists a constant C > 0 (independent of τ m ) such that u m ≤ Cτ m in K.

Proof of Corollary 5.2

We argue by contradiction. Choose d such that

P ∈K Q - d (P ) {t ≤ 0} ⊂⊂ Q - R ∩ {u ∞ = 0}. Suppose that u m (P m ) ≥ C m τ m for P m ∈ K and C m → ∞.
Clearly, for m large enough we have

λ m = C 0 R n+2 p d 2-n+2 p 1 |Q - R | Q - R |f m (x, t) -f m (0)| p 1 p < C m τ m 2 ,
and thus u m (P m ) > 2λ m .

Then, from Proposition 5.1, we know that sup

Q - d (Pm) u m ≥ d 2 2n + 1 , which implies sup Q - d (P∞) u ∞ ≥ d 2 2n + 1
,

where P m → P ∞ ∈ K. This is a contradiction.

A compactness result

The main result of this section will be Corollary 6.7 which shows the compactness in L p of certain sequences. This result will be applied in the next section.

Lemma 6.1 (Cacciopoli type estimate)

Let u be a solution of (6.1)

   Hu = f χ {u>0} u ≥ 0 in Q - 1 , f (0) ≥ 0,
and P be a solution of (6.1) with f replaced by the constant function f (0). Furthermore, set w = u -P and W = w|w|

∈ C ∞ 0 (R n+1 ) such that supp η ⊂ Q r with 0 < r ≤ 1, we have (6.2) Q - 1 2(p -1) p 2 |∇W | 2 η 2 + f (0)(χ {u>0} -χ {P >0} )|w| p-2 wη 2 + B 1 ×{0} η 2 W 2 p ≤ Q - 1 2 p -1 W 2 |∇η| 2 + 2 p |ηη t |W 2 + ω(r)|Q - r | 1 |Q - r | Q - r η 2p ′ W 2 1 p ′ .
Proof of Lemma 6.1

Recall that w = u -P solves the equation

Hw = f χ {u>0} -f (0)χ {P >0} in Q - 1 .
Multiplying this equation with η 2 w|w| p-2 we get

Q - 1 (Hw)η 2 w|w| p-2 = Q - 1 f (0)(χ {u>0} -χ {P >0} )w|w| p-2 η 2 + (f -f (0))χ {u>0} w|w| p-2 η 2 .
Furthermore,

Q - 1 (Hw)η 2 w|w| p-2 = Q - 1 η 2 w|w| p-2 ∆w + 2ηη t |w| p p - Q - 1 ∩{t=0} η 2 |w| p p , and
Q - 1 -η 2 w|w| p-2 ∆w = Q - 1 4(p -1) p 2 η 2 |∇W | 2 + 4 p ηW ∇η∇W.
Therefore, with λ = 4(p-1)

p 2
we have

Q - 1 λη 2 |∇W | 2 + f (0)(χ {u>0} -χ {P >0} )|w| p-2 wη 2 + Q - 1 ∩{t=0} η 2 W 2 p ≤ Q - 1 - 4 p ηW ∇η∇W + 2ηη t W 2 p + η 2 |f -f (0)|W 2(p-1) p ≤ Q - 1 1 2 λη 2 |∇W | 2 + 16 p 2 λ -1 W 2 |∇η| 2 + 2ηη t W 2 p + ω(r)|Q - r | 1 |Q - r | Q - r η 2p ′ W 2 1 p ′ .
This implies

Q - 1 1 2 λη 2 |∇W | 2 + f (0)(χ {u>0} -χ {P >0} )|w| p-2 wη 2 + Q - 1 ∩{t=0} η 2 W 2 p ≤ Q - 1 8 p 2 λ -1 W 2 |∇η| 2 + 2|ηη t | W 2 p + ω(r)|Q - r | 1 |Q - r | Q - r η 2p ′ W 2 1 p ′
, which is the desired inequality.

Lemma 6.2 (L ∞ t L 2
x -estimates for W ) Under the same assumptions as in Lemma 6.1, we have for any

t 0 ∈ [-1 4 , 0] Q - 1 2 ∩{t=t 0 } W 2 ≤ C Q - 1 W 2 + (ω(1)) p .
Proof of Lemma 6.2 Let w t 0 (x, t) = w(x, t -t 0 ) with u t 0 (x, t) = u(x, t -t 0 ) and P t 0 (x, t) = P (x, t -t 0 ). Then we have

Hw t 0 = f t 0 χ {u t 0 >0} -f (0)χ {P t 0 >0} in Q -√ 1-t 2 0 . Put W t 0 = W (x, t -t 0 ) and take η ∈ C ∞ 0 (R n+1 ) such that supp η ⊂ Q r with r > 1/2 and 1/4 + r 2 < 1 (so that Q - r ⊂ Q -√ 1-t 2 0
), and η = 1 on B 1 2 × {t = 0}. We now apply the proof of Lemma 6.1 to W t 0 together with Young's inequality applied to the last term of (6.2). This gives

Q - 1 ∩{t=0} (W t 0 ) 2 η 2 ≤ C Q - r (W t 0 ) 2 + Q - r |f t 0 -f (0)| p . Since Q - r (W t 0 ) 2 ≤ Q - 1 W 2 , Q - r |f t 0 -f (0)| p ≤ Q - 1 |f -f (0)| p = |Q - 1 |(ω(1)) p and Q - 1 2 ∩{t=t 0 } W 2 ≤ Q - 1 2 ∩{t=0} (W t 0 ) 2 η 2 ,
this implies the result.

Corollary 6.3 (L q t W 1,q x -estimates for w) Under the same assumptions as in Lemma 6.1, there is q ∈ (1, p] such that for any 0 < r < 1, we have

Q - r |∇w| q ≤ C(p, r, ω(1), ||w|| L p (Q - 1 )
).

Proof of Corollary 6.3

We divide the proof into two cases, depending on whether p > 2 or not. In order to clarify the dependence of ω on p, we write it ω p . Case 1: p ≥ 2. In this case we can simply apply Lemma 6.1 with p = 2 to obtain for 0 < r < 1:

Q - r |∇w| 2 ≤ C(r, ω 2 (1), ||w|| L 2 (Q - 1 ) ) ≤ C(r, ω p (1), ||w|| L p (Q - 1 )
) Case 2: p < 2. We compute for 0 < r < 1:

Q - r |∇w| q = Q - r |∇w| q |w| q(p/2-1) |w| -q(p/2-1) ≤ Q - r |∇w| 2 w p-2 q/2 Q - r (|w| -q(p/2-1) ) (2/q) ′ 1/(2/q) ′ ≤ C(r, ω p (1), ||w|| L p (Q - r ) )
The first factor can be estimated using Lemma 6.1 and the second one by C||w|| L p (Q - r ) . This is due to the fact that (2/q) ′ = (1 -q/2) -1 which implies that the exponent of w is -q(p/2 -1)/(1 -q/2) = -q(p -2)/(2 -q). We realize that for q = 1 this equals 2 -p < p, and hence if we take q > 1 small enough the exponent will be less than p. Lemma 6.4 (Partial L 1 -estimates of the right hand side) Under the same assumptions as in Lemma 6.1, for any 0 < r < 1 there holds

(6.3) Q - r f (0)|(χ {u>0} -χ {P >0} )| ≤ C(p, r, ω(1), ||w|| L p (Q - 1 )
).

Proof of Lemma 6.4

For ε ≥ 0 define

sgn ε (w) =    1 if w > ε, w ε if |w| ≤ ε, -1 if w < -ε, and h ε = f (0)(χ {u>0} -χ {P >0} ) sgn ε (w). As before we have Hw = (f -f (0))χ {u>0} + f (0)(χ {u>0} -χ {P >0} ). Take η ∈ C ∞ 0 (Q - ρ ) with η = 1 on Q - r for r < ρ < 1. Multiplying the equation by sgn ε (w)η 2 yields Q - 1 h ε η 2 = - Q - 1 (f -f (0))χ {u>0} η 2 sgn ε (w) + Q - 1 (Hw) sgn ε (w)η 2 .
We observe that sgn ε (w) is the derivative of a convex function β ε (w) ≥ 0. Therefore

Q - 1 (Hw) sgn ε (w)η 2 = Q - 1 η 2 sgn ε (w)∆w + 2ηη t β ε (w) - Q - 1 ∩{t=0} η 2 β ε (w),
and -

Q - 1 η 2 sgn ε (w)∆w = Q - 1 |∇w| 2 1 ε χ {|w|≤ε} η 2 + 2η∇η∇w sgn ε (w).
Adding up, this gives

Q - 1 h ε η 2 + Q - 1 ∩{t=0} η 2 β ε (w) + Q - 1 |∇w| 2 1 ε χ {|w|≤ε} η 2 ≤ Q - 1 |f -f (0)|η 2 + 2|ηη t |β ε (w) + 2η|∇η∇w|.
Observing that |β ε (w) -|w|| ≤ ε/2, we see

Q - r h ε ≤ Cω(1) + C(r) ||w|| L 1 (Q - 1 ) + ε + ||∇w|| L 1 (Q - ρ ) .
Hence, the Dominated convergence theorem implies

h ε → h 0 = f (0) χ {u>0} -χ {P >0} sgn(w) in L 1 (Q - r )
. Thus, by Corollary 6.3,

Q - r h 0 ≤ C(p, r, ω(1), ||w|| L p (Q 1 ) ),
which gives (6.3). This ends the proof of the lemma.

Below we state a result of Simon we will be using (see Theorem 6 on page 86 in [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]) and a small lemma that we will need. Theorem 6.5 (Compactness in Banach spaces) Let X 0 ⊂ X ⊂ X 1 be Banach spaces such that X 0 is compactly embedded in X and X is continuously embedded in X 1 . Moreover, assume that u k is a sequence of functions such that for some q > 1

||u k || L q (I;X) + ||u k || L 1 (I;X 0 ) + ||∂ t u k || L 1 (I;X 1 ) ≤ C,
where I ⊂ R is a compact interval. Then there is a subsequence u k j that converges in L p (I; X) for all 1 ≤ p < q.

Lemma 6.6 (Inclusion in dual spaces) For any r there holds

∇L s (B r ) ⊂ W -1,s (B r ) and L 1 (B r ) ⊂ W -1,s (B r ), whenever 1 < s < n n -1 .
Proof of Lemma 6.6

We first we prove the inclusion

∇L s (B r ) ⊂ W -1,s (B r ),
where

W -1,s (B r ) is by definition the dual of the space W 1,s ′ 0 (B r ) with 1/s + 1/s ′ = 1. Let f ∈ L s (B r ) and φ ∈ C ∞ 0 (B r ).
Then, by Hölder's inequality

| ∇f, φ | = | f, ∇φ | ≤ ||f || L s (Br ) ||φ|| W 1,s ′ 0 (Br) . Since C ∞ 0 (B r ) is dense in W 1,s ′ (B r ), this implies ||∇f || W -1,s (Br) = ||∇f || (W 1,s ′ 0 (Br )) * ≤ ||f || L s (Br) .
Now we prove that L 1 (B r ) ⊂ W -1,s (B r ) when s < n n-1 . In order to do so, take g ∈ L 1 (B r ) and φ ∈ W 1,s ′ 0 . Hölder's inequality implies

Br gφ ≤ ||g|| L 1 (Br) ||φ|| L ∞ (Br ) ≤ ||g|| L 1 (Br) ||φ|| W 1,s ′ (Br ) ,
whenever s ′ > n, which is equivalent to s < n n-1 . This ends the proof of the lemma.

Combining these two results with the previous section, we can conclude to the following compactness result.

   Hu k = f k χ {u k >0} u k ≥ 0 in Q - 1 , f k (0) ≥ 0,
and

HP k = f k (0)χ {P k >0} P k ≥ 0 in Q - 1 .
Assume further that with w k = u k -P k there holds for some p ∈ (1, +∞) (6.4)

||w k || L p (Q - 1 ) + ω(f k , 1) ≤ C.
Then there is a subsequence of w k converging in L p (Q -

). Moreover there is q > 1 such that (6.5)

||w k || L q t (W 1,q x )(Q - 1 2
) ≤ C.

Proof of Corollary 6.7

The proof is divided into two parts. Part 1: (Convergence a.e.) From Lemma 6.2 and Corollary 6.3 it follows that (for q > 1)

(6.6) ||w k || L ∞ t (L p x )(Q - 1 2 ) , ||∇w k || L q t (L q x )(Q - 1 2
) ≤ C.

Moreover,

∂ t w k = div(∇w k ) + f k (0)(χ {u k >0} -χ {P k >0} ) + (f k -f k (0))χ {u k >0} = a k + b k + c k .
By Lemma 6.6, up to reducing q < n n-1 , we have ∇L q (B r ) ⊂ W -1,q (B r ) for any B r . Then Corollary 6.3, Lemma 6.4 and (6.4) imply that

||a k || L 1 t (W -1,q x )(Q - 1 2 ) , ||b k || L 1 (Q - 1 2 ) , ||c k || L 1 (Q - 1 2
) ≤ C. Lemma 6.6 also implies that L 1 (B r ) ⊂ W -1,q (B r ) for any B r . Therefore

||∂ t w k || L 1 t (W -1,q x )(Q - 1 2
) ≤ C ′ . Now we wish to apply Theorem 6.5. Let X 0 = W 1,q (B 1 2

), X = L 1 (B 1 2
) and X 1 = W -1,q (B 1

2

). We can then conclude that there is a subsequence of w k that converges in

L 1 t (L 1 x )(Q - 1 2 ) = L 1 (Q - 1 2
). Hence, there is a subsequence that converges a.e..

Part 2: (L p -convergence)

We wish to apply Theorem 6.5 to the sequence |w k | p . In order to be able to do that, we need estimates.

Step A: Bound on

∇|w k | p We set W k = w k |w k | p 2 -1 .
We observe that from Lemma 6.2 and (6.4), we have

(6.7) ||W k || L ∞ t (L 2 x )(Q - 1 2
) ≤ C 1 .

On the other hand, Lemma 6.1 implies (6.8)

||∇W k || L 2 t (L 2 x )(Q - 1 2
) ≤ C 2 .

Therefore, from the elliptic Sobolev embedding (with an abuse of notation if n = 1) (6.9)

||W k || L 2 t (L 2n n-2 x )(Q - 1 2
)

≤ C 3 .
Then the interpolation between (6.7) and (6.9) gives for any α ∈ (0, 1) (this is a classical result which can for instance be easily deduced from the L p -interpolation in Brezis [START_REF] Brezis | Analyse fonctionnelle, Collection Mathematiques Appliquees pour la Maitrise[END_REF] page 57)

||W k || L p α t t (L p α x x )(Q - 1 2
)

≤ C 4 , with 1 p α t = 1 -α ∞ + α 2 < 1 2 , 1 p α x = 1 -α 2 + α 2n n-2 < 1 2 .
This implies the existence of some p 0 > 2 such that (6.10)

||W k || L p 0 (Q - 1 2
) ≤ C 5 .

Then, with the q given in Part 1

1 2 q Q - 1 2 |∇|w k | p | q = 1 2 q Q - 1 2 |2W k ∇W k | q ≤   Q - 1 2 |∇W k | 2   q 2   Q - 1 2 |W k | q( 2 q ) ′   1 ( 2 q ) ′ .
But q( 2 q ) ′ = 2q 2-q is increasing in q with value 2 for q = 1. Therefore, under our assumptions, and up to reducing q > 1, we can chose q such that q( 2 q ) ′ ≤ p 0 . We then use (6.8) and (6.10) to conclude that (6.11)

||∇|w k | p || L q (Q - 1 2
) ≤ C 6 .

Step B: Using the PDE to conclude In order to obtain some information about ∂ t |w k | p we need to play with the equation for w k again. Multiplication by

|w k | p-1 sgn(w k ) gives -∂ t |w k | p p + |w k | p-1 sgn(w k )∆w = f k (0)(χ {u k >0} -χ {P k >0} ) + (f k -f k (0))χ {u k >0} |w k | p-1 sgn(w k ).
Rearranging a bit this yields

-∂ t |w k | p p = f k (0)(χ {u k >0} -χ {P k >0} ) + (f k -f k (0))χ {u k >0} |w k | p-1 sgn(w k ) -div((|w k | p-1 sgn w k ∇w k ) +(p -1)|w k | p-2 |∇w k | 2 = a k + b k + c k .
An observant reader might see that a priori, the calculations above are not valid other than in some formal sense. However, a simple approximation argument can make this rigorously justified. Now, Lemma 6.1, the bound (6.4) and Hölder's inequality imply that (6.12)

||a k || L 1 (Q - 1 2 ) , ||c k || L 1 (Q - 1 2
) ≤ C.

Estimate (6.11) and Lemma 6.6 imply

||b k || L 1 t (W -1,q x )(Q - 1 2
)

and estimate (6.12) and Lemma 6.6 imply

||a k || L 1 t (W -1,q x )(Q - 1 2 ) , ||c k || L 1 t (W -1,q x )(Q - 1 2
) ≤ C.

Hence,

||∂ t |w k | p || L 1 t (W -1,q x )(Q - 1 2
) ≤ C.

Applying Theorem 6.5 for the sequence |w

k | p with X 0 = W 1,q (B 1 2 ), X = L 1 (B 1
2

) and X 1 = W -1,q (B 1 2 ), we find that, up to a subsequence, |w k | p converges in L 1 . This, together with the a.e.-convergence of a subsequence of w k , implies the existence of a subsequence of w k converging in L p (Q - ).

Part 3: (Proof of (6.5)) Finally, (6.5) follows from Corollary 6.3 and the bound (6.4). This ends the proof of the corollary.

7 Decay estimates and the proof of Theorem 1.7

The aim of this section is to prove Theorem 1.7. The key result is Proposition 7.2. We define

N(u, ρ) = inf P ∈Preg 1 ρ n+2+2p Q - ρ |u -P | p dx dt 1 p , and M(u, r) = sup 0<ρ≤r N(u, ρ),
which is nothing else than the quantity M reg (u, r) defined in the introduction. We will need the following result:

Lemma 7.1 (Estimates of N in larger balls) If N(u, 1) = ||u -P 1 || L p (Q - 1 ) , with P 1 ∈ P reg , then for any ρ ≥ 1, 1 ρ n+2+2p Q - ρ |u -P 1 | p dx dt 1 p ≤ C 1 2ρ 1 N(u, s) s ds.
Proof of Lemma 7.1

The proof of this lemma is similar to the proof of Lemma 2.9 in [START_REF] Monneau | Pointwise estimates for Laplace equation. Applications to the free boundary of the obstacle problem with Dini coefficients[END_REF], which is proved by decomposing B ρ into dyadic balls and estimating the quantities in each of the balls. We notice in particular that for α ∈ [1, 2], we have

N(u, αρ) ≤ 2 2+ n+2 p N(u, 2ρ),
which is used in order to get the result with the integral of N on the right hand side.

Proposition 7.2 (Decay estimate)

Let u be a solution of (1.6). Then there are constants M 0 , C 0 > 0, r 0 , λ, µ ∈ (0, 1) such that for all r < r 0

M(u, r) ≤ M 0 =⇒ M(u, λr) < µM(u, r) or M(u, r) < C 0 σ(f, r).
Proof of Proposition 7.2

Step 1: Construction of sequences and a priori estimates We argue by contradiction. If this is not true, we can find C k → ∞, M k , r k , λ k → 0 and µ k → 1 such that the statement above fails with the corresponding functions u k and f k , i.e., we have

M(u k , r k ) ≤ M k but still M(u k , λ k r k ) ≥ µ k M(u k , r k ) and M(u k , r k ) ≥ C k σ(f k , r k ).
We note that by our assumption, M(u k , λ k r k ) → 0. This implies that we can, passing to another subsequence if possible, assume that for some 0

< ρ k ≤ λ k r k M(u k , λ k r k ) 1 + 1/k ≤ N(u k , ρ k ) = ε k → 0.
Define the rescaled functions

v k (x, t) = u k (ρ k x, ρ 2 k t) ρ 2 k and (7.1) w k (x, t) = u k (ρ k x, ρ 2 k t) -P k (ρ k x) ε k ρ 2 k ,
where P k ∈ P reg is a half-space function realizing the infimum defining N(u k , •) at the level ρ k . Now, we wish to pass to the limit, but before doing that we need to gather up some informative estimates mainly on the functions w k . By definition (7.2) inf

P ∈Preg Q - 1 w k - P -P k ε k p dx dt 1 p = 1.
Moreover, with sρ k ≤ r k , we have inf

P ∈Preg 1 s n+2+2p Q - s w k - P -P k ε k p dx dt 1 p = N(u k , sρ k ) ε k ≤ M(u k , r k ) ε k ≤ M(u k , λ k r k ) ε k µ k ≤ 1 + 1/k µ k . (7.3) Since N(v k , 1) = Q - 1 |v k -P k | p dx dt 1 p
, we also have for s ∈ (1, r k 2ρ k ) (7.4)

1 s n+2+2p Q - s |w k | p dx dt 1 p = 1 ε k 1 s n+2+2p Q - s |v k -P k | p dx dt 1 p ≤ C 1 ε k 2s 1 M(v k , τ ) τ dτ = C 1 ε k 2s 1 M(u k , τ ρ k ) τ dτ ≤ C 1 ε k 2s 1 M(u k , r k ) τ dτ ≤ C 1 ε k 1 + 1/k µ k N(u k , ρ k ) ln 2s ≤ C 2 ln 2s,
where, in the second line we have used Lemma 7.1. Up to rotating our coordinates we can assume that P k (x, t) = 1 2 (max(0, x 1 )) 2 .

Then

Hw k = g k in {v k > 0} ∩ {x 1 > 0}, with g k (x, t) = f k (ρ k x, ρ 2 k t) -f k (0) ε k and f k (0) = 1.
Furthermore, for any 0 < r < r k ρ k → ∞ as k → ∞, there holds

1 |Q - r | Q - r |g k | p dx dt 1 p ≤ σ(f k , ρ k r) ε k ≤ M(u k , r k ) C k ε k ≤ M(u k , λ k r k ) µ k C k ε k ≤ 1 + 1/k µ k C k → 0.
Step 2: Passing to the limit Corollary 6.7 implies that, up to a subsequence, the sequence w k converges to some function w ∞ in L p loc . From the L p -bound (7.4) on w k and (7.1) it follows that

v k → P ∞ = 1 2 (max(x 1 , 0)) 2 in L p loc .
From (7.4) we deduce that for any s > 1 there holds Moreover, (7.3) gives that for any s > 0, (7.7) inf

(7.5) 1 s n+2+2p Q - s |w ∞ | p dx dt
q∈T P∞ Preg 1 s n+2+2p Q - s |w ∞ -q| p dx dt 1 p ≤ 1.
From the equation for w k it follows that (7.8) Hw ∞ = 0 in {P ∞ > 0} = {x 1 > 0}.

In addition

Hv k = f k (ρ k x, ρ 2 k x)χ {v k >0}
, where the right hand side is bounded in L p loc . Therefore, by the interior estimates and the Sobolev embedding (Theorem 2.1 and Theorem 2.2), for a subsequence v k → P ∞ in L ∞ loc . Moreover, the weak non-degeneracy (Corollary 5.2) implies that for any compact

K ⊂ {x 1 < 0} ∩ Q - R , we have v k ≤ Cσ(f k , Rρ k ) in K.
Thus,

w k ≤ Cσ(f k , Rρ k ) ε k = Cσ(g k , R) → 0 in K,
as k → ∞, and then (7.9) w ∞ = 0 in {x 1 < 0}.

Finally, from (6.5) in Corollary 6.7, we also deduce w ∞ ∈ (L q t (W 1,q x )) loc for some q > 1.

Step 3: Identification of the limit and contradiction Since {x 1 = 0} is of codimension one in space and does not depend on time, there is a trace of w ∞ , enjoying the estimate

||w ∞ || L q (Q - r ∩{x 1 =0}) ≤ C||w ∞ || L q t (W 1,q x )(Q - r ∩{x 1 <0}
) , for any r > 0. From (7.9), we deduce that w ∞ = 0 on {x 1 = 0}. Define w * to be the odd reflection of w ∞ with respect to the plane {x 1 = 0}. Due to (7.8), w * is caloric in R n × (-∞, 0). Moreover, by (7.5), w ∞ grows strictly slower than r 3 at infinity. This implies that w * is a caloric polynomial, and w * (x, t) = αt + P (x), where P is a polynomial of degree less than or equal to two. Taking s → 0 in (7.7), we see that P must be homogeneous of degree two. Indeed, any linear (spatial) or constant part of w ∞ would make the right hand side of (7.7) blow up as s → 0. Since w * vanishes on {x 1 = 0}, we have α = 0 and w * must be a spatial harmonic polynomial homogeneous of degree two. Therefore, as in [START_REF] Monneau | Pointwise estimates for Laplace equation. Applications to the free boundary of the obstacle problem with Dini coefficients[END_REF] (Step 3 of the proof of Proposition 6.2), we see that w * ∈ T P∞ P reg which contradicts (7.6). This ends the proof of the proposition.

Proof of Theorem 1.7

The theorem follows by combining Proposition 3.2 in [START_REF] Monneau | Pointwise estimates for Laplace equation. Applications to the free boundary of the obstacle problem with Dini coefficients[END_REF] with the present Proposition 7.2.

1 σDefinition 1 . 1 (

 111 (r) = sup |x-y| 2 +|t-s|≤r 2 (x,t),(y,s)∈Q - 1 |f (x, t) -f (y, s)|. Dini function) A function σ is said to be Dini if

  VMO estimate ω(r) -→ 0 as r → 0 + =⇒ Ñ(u, r) -→ 0 as r → 0 +

  and P 0 (x, t) = a + b • x + 1 2 t x • c • x + mt, with |a| + |b| + |c| + |m| ≤ C M0 and M0 =

Remark 3 . 4

 34 Notice that the quantities N and ω defined in (3.12) satisfy (3.13) because of the basic estimate (Proposition 3.1) and do also satisfy(3.14) because of (3.11) with γ = 1/λ (with

Theorem 1. 2

 2 Proof of Theorem 1.2 Proof of i) Using definition (3.12) of N and ω, and estimate (3.15) with the constant C 2 given in (3.16), we deduce that for r ∈ (0, λ] N (r) ≤ C 0 N (1) + sup ρ∈(0,1] ω(ρ) . From (3.11) with γ = 1/λ, we deduce that for all r ∈ (0, 1] N(u, r) ≤ C N(u, 1) + sup ρ∈(0,1] ω(f, ρ) which implies (1.4) because we always have Ñ (u, r) ≤ N (u, r) and N (u, 1)

p 2 - 1

 21 for p ∈ (1, +∞). Then for any η

Corollary 6 .

 6 7 (Compactness) Assume we have sequences of functions u k and P k such that

1 p≤ C 2 ( 7

 127 ln 2s.We define the tangent space of P reg at P ∞ asT P∞ P reg = x + 1 (x, β) where β = (β 1 , . . . , β n ) ∈ R n and β 1 = 0 , which is exactly the set of all possible limits of Pk -P ∞ ε k ,as k → ∞, with ε k → 0 and Pk → P ∞ . Then from (7.2) together with the local L pconvergence of w k

Acknowledgements

Both of the authors have partially been supported by the ANR project "MICA", grant ANR-08-BLAN-0082. Moreover, the first author has been partially supported by the Chair "Mathematical modelling and numerical simulation, F-EADS -Ecole Polytechnique -IN-RIA", NTNU and MSRI. The authors also thank the MSRI for providing excellent conditions of work and they are very grateful to Peter Lindqvist and Giuseppe Mingione, for the help with finding proper references.


Andrew Om Wilkie
email: awilkie@hammer.imm.ox.ac.uk

Jane A Hurst

Dagan Jenkins

Pradeep C Vasudevan

Maria Kirchhoff

Flemming Skovby

Claudine Rieubland

Sabina Gallati

Olaf Rittinger

David Johnson

Peter M Kroisel

Leslie G Biesecker

Metopic and sagittal synostosis in Greig cephalopolysyndactyly syndrome: five cases with intragenic mutations or complete deletions of GLI3

Keywords: trigonocephaly, metopic synostosis, sagittal synostosis, Greig cephalopolysyndactyly syndrome, GLI3, Carpenter syndrome

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In 1926, Greig [START_REF] Greig | Oxycephaly[END_REF] first described the condition now known as Greig cephalopolysyndactyly syndrome (GCPS, MIM 175700) in a mother and daughter with digital malformations and an unusual skull shape. The syndrome is inherited in an autosomal dominant pattern and presents with characteristic limb and craniofacial anomalies. It is highly penetrant but there is both inter-and intra-familial variability of the features. [START_REF] Fryns | The Greig polysyndactyly craniofacial dysmorphism syndrome: variable expression in a family[END_REF][START_REF] Baraitser | Greig cephalopolysyndactyly: report of 13 affected individuals in three families[END_REF] In the extremities there may be both preaxial and postaxial polydactyly with broad or duplicated thumbs, broad or duplicated halluces, postaxial polydactyly of hands and feet and cutaneous syndactyly of the fingers or toes.

Craniofacial features include macrocephaly, prominent or broad forehead with frontal bossing, a broad base to the nose and a wide nasal bridge. Some affected individuals have learning difficulties. Developmental anomalies of the corpus callosum and mild cerebral ventriculomegaly are recognised associations. To our knowledge, trigonocephaly was described only twice in possible GCPS cases before molecular confirmation became available. [START_REF] Hootnick | Familial polysyndactyly and craniofacial anomalies[END_REF][START_REF] Guzzetta | Polysyndactyly and trigonocephaly with partial agenesis of corpus callosum[END_REF] Following the recognition that GCPS was associated with rearrangements and translocations of 7p14.1, [START_REF] Tommerup | A familial reciprocal translocation t(3;7)(p21.1;p13) associated with the Greig polysyndactyly-craniofacial anomalies syndrome[END_REF][START_REF] Brueton | Chromosomal localisation of a developmental gene in man: direct DNA analysis demonstrates that Greig cephalopolysyndactyly maps to 7p13[END_REF][START_REF] Drabkin | Regional and physical mapping studies characterizing the Greig polysyndactyly 3;7 chromosome translocation[END_REF][START_REF] Pettigrew | Greig syndrome associated with an interstitial deletion of 7p: confirmation of the localization of Greig syndrome to 7p13[END_REF] heterozygous GLI3 mutations and deletions were identified as the cause of GCPS by Vortkamp et al in 1991. [START_REF] Vortkamp | GLI3 zinc-finger gene interrupted by translocations in Greig syndrome families[END_REF][START_REF] Vortkamp | Deletion of GLI3 supports the homology of the human Greig cephalopolysyndactyly syndrome (GCPS) and the mouse mutant extra toes (Xt)[END_REF] The GLI3 gene encodes a zinc finger transcription factor that is a downstream mediator of the sonic hedgehog (SHH) pathway. In the presence of SHH, full length GLI3 acts as a transcriptional activator, whereas in the absence of SHH, GLI3 is cleaved to produce a repressor. [START_REF] Biesecker | What you can learn from one gene: GLI3[END_REF] The phenotype of GCPS is attributable to functional haploinsufficiency of GLI3; when this is caused by heterozygous deletion at 7p14.1, additional developmental delay may be attributable to the deletion of contiguous gene(s). [START_REF] Kroisel | Phenotype of five patients with Greig syndrome and microdeletion of 7p13[END_REF][START_REF] Johnston | Clinical and molecular delineation of the Greig cephalopolysyndactyly contiguous gene deletion syndrome and its distinction from acrocallosal syndrome[END_REF] Here we describe five unrelated individuals (four boys and a girl) with GLI3 mutation-proven GCPS and a typical pattern of pre-and post-axial polydactyly, all of whom had trigonocephaly caused by metopic synostosis, confirming that this is a nonrandom association. The key clinical features and results of molecular genetic analyses are summarised in Table 1.

Case reports

Patient 1 This boy (Figure 1a-c), the third child of healthy unrelated parents, was tabulated (subject G31) in the report by Johnston et al. [START_REF] Johnston | Molecular analysis expands the spectrum of phenotypes associated with GLI3 mutations[END_REF] He was born at term by normal vaginal delivery weighing 4132 g (95 th centile) and noted to have a prominent metopic ridge. At 10.5 months his occipito-frontal circumference (OFC) was 48 cm (90th centile) and inner canthal distance 2.9 cm (95 th centile). He had bilateral postaxial polydactyly type B of the fingers and preaxial polydactyly of the feet, with cutaneous syndactyly between the hallux and second toe and partial cutaneous syndactyly of second and third toes. Cranial computed tomography (CT) scan showed metopic and sagittal synostosis. He attained his developmental milestones normally.

Corrective surgery was performed on the hands and feet at 10 months and 1 year, respectively, and trigonocephaly correction at the age of 2 years.

DNA sequencing showed a heterozygous nonsense mutation in exon 10 of GLI3 (c.1728C>A, p.Y576X), [START_REF] Johnston | Molecular analysis expands the spectrum of phenotypes associated with GLI3 mutations[END_REF] which was not present in either parent. This mutation is predicted to truncate GLI3 in the zinc finger motif and falls within the mutation spectrum previously described as associated with GCPS. [START_REF] Johnston | Molecular analysis expands the spectrum of phenotypes associated with GLI3 mutations[END_REF][START_REF] Johnston | Molecular and clinical analyses of Greig cephalopolysyndactyly and Pallister-Hall syndromes: robust phenotype prediction from the type and position of GLI3 mutations[END_REF] Patient 2 This boy (Figure 1d-f) was included in the series of patients with metopic synostosis reported by Kini et al. [START_REF] Kini | Etiological heterogeneity and clinical characteristics of metopic synostosis: evidence from a tertiary craniofacial unit[END_REF] He was referred for craniofacial assessment at 3 years with trigonocephaly. He had healthy unrelated white British parents. He was born at term weighing 3260 g (25th centile), and was noted to have limb malformations and a trigonocephalic appearance. He had partially duplicated thumbs, bilateral postaxial polydactyly type B of the hands and bilateral preaxial polydactyly of the halluces with cutaneous syndactyly between the duplicated halluces and the second and third toes, which have required multiple corrective surgical procedures. At 4.2 years his OFC was 52 cm (75th centile) and developmental milestones were normal. Three-dimensional cranial CT scanning demonstrated metopic synostosis. He underwent fronto-orbital advancement and remodelling at the age of 4.5 years because of parental concerns about his appearance.

DNA sequencing showed a heterozygous de novo 1 bp insertion after nucleotide 1793 in exon 11 of GLI3, causing a frameshift [c.1793dupA, p.N598KfsX7], which was not present in either parent. Similar to patient 1, this mutation is predicted to truncate GLI3 within the zinc finger motif.

Patient 3 This girl (Figure 1g-i) was born at term to unrelated parents, with a birth weight of 2440 g (<3 rd centile), length 48 cm (10 th centile) and OFC 32.8 cm (3 rd centile). She had broad thumbs and halluces, partial cutaneous syndactyly between the first and second toes bilaterally and between the second and third toes on the right, and a trigonocephalic shape to the forehead. A cranial CT scan confirmed metopic synostosis. Additional features were a ventricular septal defect (VSD) which closed spontaneously, moderate developmental delay (see Table 1 for details), and dysmorphic facial features (narrow palpebral fissures, a broad nasal bridge). At 15 years she was 160 cm tall (50th centile) and weight was 68 kg (91 st centile). The karyotype was normal (46,XX).

A clinical diagnosis of Carpenter syndrome [START_REF] Jenkins | RAB23 mutations in Carpenter syndrome imply an unexpected role for hedgehog signaling in cranial-suture development and obesity[END_REF][START_REF] Jenkins | Carpenter syndrome: extended RAB23 mutation spectrum and analysis of nonsense-mediated mRNA decay[END_REF] he did not sit until 2 years of age and at 3.5 years was able to walk with one hand held. He had a vocabulary of about 15 words but had better receptive language skills.

The karyotype was normal (46,XY).

A clinical diagnosis of Carpenter syndrome was proposed, but in view of the similar clinical features to previous cases, aCGH was performed. This demonstrated a 6.0 Mb deletion (39,213,707-45,251,621, hg18 build) at 7p13-p14.1 encompassing GLI3 and 51 other RefSeq genes, which was not present in either parent (Figure 2c).

Discussion

This series of five patients with GLI3 mutations associated with synostosis of the midline cranial sutures (metopic in all five cases and additional sagittal synostosis in two cases), establishes this as an important, low frequency association of GCPS. It confirms and extends the recent reports of McDonald-McGinn et al, [START_REF] Mcdonald-Mcginn | Metopic craniosynostosis due to mutations in GLI3: a novel association[END_REF] who described two cases of metopic synostosis associated with intragenic truncating mutations of GLI3, and Johnston et al, [START_REF] Johnston | Molecular analysis expands the spectrum of phenotypes associated with GLI3 mutations[END_REF] which tabulated (but did not comment on) three GLI3 mutation positive cases with craniosynostosis -comprising Patient 1 presented here, one subject (G5-2, frameshift mutation) with metopic synostosis, and one (G34-2, missense mutation) with sagittal synostosis associated with agenesis of the corpus callosum and mild ventricular prominence.

Mutations in GLI3, which contains 15 exons and spans 280 kb, have been described not only in GCPS but also in Pallister-Hall syndrome (PHS), in preaxial polydactyly type IV, and in one patient previously diagnosed as acrocallosal syndrome. [START_REF] Elson | De novo GLI3 mutation in acrocallosal syndrome: broadening the phenotypic spectrum of GLI3 defects and overlap with murine models[END_REF][START_REF] Cohen | Editorial: perspectives on craniosynostosis[END_REF] Although GCPS and PHS share some overlapping features (notably polydactyly), they are usually readily distinguished clinically. The distinction between GCPS and preaxial polydactyly type IV is based on the absence of any craniofacial features or learning difficulties in the latter condition. GCPS is caused by inactivating mutations and deletions [START_REF] Kroisel | Phenotype of five patients with Greig syndrome and microdeletion of 7p13[END_REF][START_REF] Johnston | Clinical and molecular delineation of the Greig cephalopolysyndactyly contiguous gene deletion syndrome and its distinction from acrocallosal syndrome[END_REF] that lead to haploinsufficiency, whereas PHS is caused by frameshift, nonsense and splicing mutations in the middle third of the gene, corresponding to exon 14 and parts of exons 13 and 15. [START_REF] Johnston | Molecular analysis expands the spectrum of phenotypes associated with GLI3 mutations[END_REF][START_REF] Johnston | Molecular and clinical analyses of Greig cephalopolysyndactyly and Pallister-Hall syndromes: robust phenotype prediction from the type and position of GLI3 mutations[END_REF] Two of the five children reported here with GCPS had intragenic GLI3 mutations causing premature truncation of the protein; these were located in exons 10 and 11, the region encoding the zinc finger domain, and these mutations are expected to cause GCPS. Other recently described GLI3 mutations associated with metopic or sagittal synostosis [START_REF] Johnston | Molecular analysis expands the spectrum of phenotypes associated with GLI3 mutations[END_REF][START_REF] Mcdonald-Mcginn | Metopic craniosynostosis due to mutations in GLI3: a novel association[END_REF] were in different exons (6, 14 or 15); both mutations reported by Johnston et al [START_REF] Johnston | Molecular analysis expands the spectrum of phenotypes associated with GLI3 mutations[END_REF] were familial, but no other mutation-positive family member (total of four individuals) had craniosynostosis. Together with the identification of three complete deletions reported here, this reinforces the conclusion that midline synostosis is a low frequency association of GCPS , rather than representing a distinct genotypic or phenotypic subgroup. The single report of GCPS (case 2) in a consecutive series of 110 individuals with metopic synostosis [START_REF] Kini | Etiological heterogeneity and clinical characteristics of metopic synostosis: evidence from a tertiary craniofacial unit[END_REF] indicates that GLI3 mutations are rare in this group of patients overall.

In children with GCPS due to large chromosomal deletions, additional clinical features may be attributable to haploinsufficiency of contiguous genes in the region of 7p14.1. Multiple genes may contribute to the more severe developmental delay in these children, and deletion of GCK (which occurred in Patients 3-5) is potentially associated with maturity onset diabetes of the young (MODY) type 2. [START_REF] Kroisel | Phenotype of five patients with Greig syndrome and microdeletion of 7p13[END_REF] In each of Patients 3-5, the combination of craniosynostosis and developmental delay had led to a presumptive diagnosis of Carpenter syndrome and referral for genetic testing.

Carpenter syndrome is an autosomal recessive condition caused by biallelic mutations in RAB23. [START_REF] Jenkins | RAB23 mutations in Carpenter syndrome imply an unexpected role for hedgehog signaling in cranial-suture development and obesity[END_REF][START_REF] Jenkins | Carpenter syndrome: extended RAB23 mutation spectrum and analysis of nonsense-mediated mRNA decay[END_REF] Compared to GCPS, craniosynostosis is a more consistent and severe component of Carpenter syndrome, but there is overlap of the polysyndactyly features. Other clinical features that would suggest Carpenter syndrome, rather than GCPS contiguous gene syndrome, are additional fusion of coronal or lambdoid sutures, high birth weight, umbilical hernia and hypogenitalism in males; these were all absent in Patients 3-5. Congenital heart disease, a recognised feature of Carpenter syndrome, was present in all three patients with GLI3 deletions. Guzzetta et al [START_REF] Guzzetta | Polysyndactyly and trigonocephaly with partial agenesis of corpus callosum[END_REF] considered both GCPS and Carpenter syndrome in the differential diagnosis of a child with polysyndactyly with trigonocephaly and partial agenesis of the corpus callosum; in the light of the new data, a GLI3 deletion may be the most likely causation. Indeed, Patient 5 exhibited hypoplasia of the corpus callosum, a feature that could also cause confusion with acrocallosal syndrome. [START_REF] Elson | De novo GLI3 mutation in acrocallosal syndrome: broadening the phenotypic spectrum of GLI3 defects and overlap with murine models[END_REF] There has been debate whether craniosynostosis is a feature of GCPS. The early report of Hootnick and Holmes 4 described trigonocephaly and metopic synostosis.

Most authors describe a broad and or prominent forehead as a consistent part of the facial gestalt, but not metopic synostosis. Craniosynostosis was mentioned as being present in 5% of patients in the first edition of Cohen's monograph on craniosynostosis, [START_REF] Cohen | Craniosynostosis: Diagnosis, Evaluation, and Management[END_REF] but in the second edition it was stated that no cases had been molecularly proven. [START_REF] Cohen Mm | Craniosynostosis: Diagnosis, Evaluation, and Management[END_REF] The current series of five patients, with the four other recently reported cases, [START_REF] Johnston | Molecular analysis expands the spectrum of phenotypes associated with GLI3 mutations[END_REF][START_REF] Mcdonald-Mcginn | Metopic craniosynostosis due to mutations in GLI3: a novel association[END_REF] now confirms that premature fusion of midline cranial sutures (metopic in 8/9 and sagittal in 3/9), is a feature of GCPS and is found in children both with intragenic mutations and contiguous deletions involving GLI3. The synostosis caused significant trigonocephaly, requiring cranial surgery in four of the five patients presented here. Whereas metopic synostosis usually leads to significant hypotelorism, this was not evident in some cases probably owing to the counteracting effect of the GLI3 mutation, which is associated with widely spaced eyes and macrocephaly. [START_REF] Fryns | The Greig polysyndactyly craniofacial dysmorphism syndrome: variable expression in a family[END_REF][START_REF] Baraitser | Greig cephalopolysyndactyly: report of 13 affected individuals in three families[END_REF] Synostosis affecting the midline sutures (both metopic and sagittal) is heterogeneous in aetiology but shares several epidemiological characteristics: [START_REF] Lajeunie | Genetic considerations in nonsyndromic midline craniosynostoses: a study of twins and their families[END_REF] several mechanisms may contribute to their association with GLI3 mutations. Fetal head constraint is a predisposing factor in non-syndromic cases, [START_REF] Lajeunie | Genetic considerations in nonsyndromic midline craniosynostoses: a study of twins and their families[END_REF][START_REF] Sanchez-Lara | Fetal constraint as a potential risk factor for craniosynostosis[END_REF] and the macrocephaly normally associated with GCPS would be expected to enhance this effect. A wide variety of chromosome abnormalities are associated with syndromic midline synostosis, suggesting that in cases with megabase-sized deletions, reduced transduction of stretch forces exerted by the growing brain may represent a nonspecific contributing factor. [START_REF] Wilkie | Prevalence and complications of single gene and chromosomal disorders in craniosynostosis[END_REF] However gene-specific mechanisms may also play a role, because mice homozygous null for the orthologous Gli3 protein have craniosynostosis, although surprisingly this affects the lambdoid sutures; by contrast the metopic and sagittal sutures are widened, with islands of ossification. [START_REF] Rice | Gli3 Xt2J/Xt2J mice exhibit lambdoid suture craniosynostosis which results from altered osteoprogenitor proliferation and differentiation[END_REF] The excess of males (four cases) over females (one case) in our series is typical of metopic synostosis as a whole. [START_REF] Kini | Etiological heterogeneity and clinical characteristics of metopic synostosis: evidence from a tertiary craniofacial unit[END_REF] In conclusion, two particular clinical management issues arise from our observations. First, all children with GCPS should be examined for evidence of trigonocephaly and/or scaphocephaly to exclude the possibility of a co-existing craniosynostosis. Second, in patients in whom a diagnosis of Carpenter syndrome is being considered, the possibility of a GLI3 deletion should be included in the differential diagnosis and specifically sought using aCGH or comparable technique.

Patient 4 Patient 5

 45 was suggested but DNA sequencing of the open reading frame of RAB23 was normal. Array comparative genomic hybridisation (aCGH; Agilent 244K array) showed an 8.3 Mb deletion at 7p12.3-p14.1 [(40845981_40855164)_(49136714_49160830); hg18 assembly],encompassing GLI3 and 59 flanking RefSeq genes, which was not present in either parent (Figure2a). This boy (Figure1j-l) was the first child born at 39 weeks to unrelated parents after a normal pregnancy, with a birth weight of 3560g (50 th centile), length of 50 cm (50 th centile), and head circumference of 34 cm (20 th centile). The hands were normal. He had bilateral duplication of the halluces and syndactyly between the second and third toes. His head shape was abnormal with a combination of scaphocephaly and trigonocephaly. Radiology confirmed synostosis of the sagittal and metopic sutures and he had surgical correction at the age of 8 months. Additional features included laryngomalacia, atrial septal defect (ASD), VSD, and patent ductus arteriosus (PDA). He developed seizures at the age of 3 years and has severe global developmental delay (Griffiths Quotient 26 aged 4 years). Ophthalmologic examination showed horizontal nystagmus and mild optic nerve hypoplasia. His brain magnetic resonance imaging and karyotype (46,XY) were normal.A clinical diagnosis of Carpenter syndrome was proposed, but in view of the similar clinical features to Patient 3, aCGH was performed. This showed a heterozygous 6.8 Mb deletion (39,130,081-45,492,392; hg18) of the region 7p13-p14.1, encompassing GLI3 and 51 flanking RefSeq genes, which was not present in either parent (not illustrated). This boy (Figure1m-o) was the first child of healthy unrelated parents. He was born at term with normal length (53 cm, 90 th centile), weight (3350 g, 50 th centile) and OFC (33.5 cm, 10 th centile). At the age of 2 weeks he was noticed to have trigonocephaly with a metopic ridge, large anterior fontanelle, low frontal hairline and upslanting palpebral fissures. X-ray confirmed premature synostosis of the metopic suture. Additional dysmorphic facial features included a flat nasal bridge and broad nasal base. His ears had poorly formed and overfolded helices. In his hands, the thumbs were proximally inserted with broadening of the terminal phalanx and he had bilateral preaxial polydactyly of the halluces with cutaneous syndactyly of the second and third toes. His cranial ultrasound demonstrated mild ventriculomegaly and a small corpus callosum. Fundoscopy was normal. Echocardiography showed a double outlet right ventricle (DORV) with pulmonary stenosis. He had delayed motor development;

Figure 1

 1 Figure 1 Clinical features of patients with trigonocephaly and mutations of GLI3. (a-

Figure 2

 2 Figure 2 Array CGH analysis of Patients 3 and 5 with deletions including GLI3. (a),

Table 1 .

 1 Clinical features and results of GLI3 testing in Patients 1-5

		Patient 1	Patient 2	Patient 3	Patient 4	Patient 5
	Sex	Male	Male	Female	Male	Male
	GLI3 testing	c.1728C>A,	c.1793dupA,	8.3 Mb deletion	6.8 Mb deletion	6.0 Mb deletion
		p.Y576X	pN598KfsX7	7p12.3-p14.1	7p13-p14.1	7p13-p14.1
	Family history	No	No	No	No	No
	Parental ages	Mother 32 yr	Mother 22 yr	Mother 25 yr	Mother 19 yr	Mother 29 yr
		Father 34 yr	Father 22 yr	Father 25 yr	Father 23 yr	Father 29 yr
	OFC	Prior to surgery: 48	Prior to surgery:	Birth: 32.8 cm (5th	Birth: 34 cm (20 th	Birth 33.5 cm (10 th
		cm at 10.5 mo (90 th	52 cm at 4 yr	centile)	centile)	centile)
		centile)	3mo
			(75 th centile)
	Trigonocephaly	Yes	Yes	Yes	Yes and	Yes
					scaphocephaly
	Craniosynostosis	Metopic and	Metopic	Metopic	Metopic and	Metopic
		sagittal			sagittal
	Structural brain	No	No	Bilateral frontal and	No	Hypoplasia corpus
	anomaly			parietal atrophy		callosum
						Mild cerebral
						ventricular
						dilatation
	Development	Normal	Normal	Delayed	Delayed	Delayed
	Sat			1 yr	14 months	2 yr
	Walked			2.5 yr	Not walking	3.5 yr
					independently at 5
					yr 9 mo
	Speech at 3 years			10-12 words	< 5 words	15 words
	Schooling			At 15 yr approx	Special schooling	-
				equivalent to
				average 7 yr
	Thumb	Normal	Partially	Broad	Normal	Broad
			duplicated
	Finger	Postaxial	Postaxial	No	No	No
	poly/syndactyly	polydactyly type B	polydactyly type
			B
	Hallux	Duplicated	Duplicated, with	Broad	Duplicated, with	Duplicated, with
		(separate preaxial	complete		partial cutaneous	partial cutaneous
		digit on left)	cutaneous		syndactyly	syndactyly
			syndactyly
	Toe cutaneous	Complete	Complete	Partial cutaneous	Partial syndactyly	Partial syndactyly
	poly/syndactyly	cutaneous	cutaneous	syndactyly between	second/third toes	second/third toes
		syndactyly hallux	syndactyly hallux	hallux/second/third
		and second toe.	and second toe.	toes
		Partial syndactyly	Partial syndactyly
		second/ third toes	second-fourth
			toes
	Other anomalies	No	No	Short palpebral	Telecanthus,	Flat nasal bridge
				fissures, flat nasal	bilateral ptosis,	and broad base.
				bridge and broad	anteverted nares,	DORV with
				base, thick lips	large mouth.	pulmonary
				Hypermetropia and	Laryngomalacia,	stenosis.
				squint. Small VSD	VSD, ASD and
				(closed	PDA
				spontaneously)	Seizures age 3
					years

Abbreviations: VSD: ventricular septal defect, ASD: atrial septal defect, PDA: patent ductus arteriosus, DORV: double outlet right ventricle

Acknowledgements

We thank all the families for their participation in this study, Sandra Bigi, Bruce Castle, Shincy John, Michael Parker and Oliver Quarrell for help with gathering clinical information and Bernard Conrad, Jennifer Johnston and Tracy Lester for assistance with molecular analysis. This work was supported by grants to AOMW from the Wellcome Trust (078666) and the Medical Research Council (80106).

Conflict of Interest