Edinburgh Explorer Lipschitz continuity of free boundary in the continuous casting problem with divergence form elliptic equation

. In this paper we are concerned with the regularity of weak solutions u to the one phase continuous casting problem in the cylindrical domain Ω × ) where X = ( x;z ) ;x ∈ Ω ⊂ R N (cid:0) 1 ;z ∈ (0 ;L ) ;L > 0 with given elliptic matrix A : Ω → R N 2 ;A ij ( x ) ∈ C 1 ,α 0 (Ω) ;(cid:11) 0 > 0, prescribed convection v , and the enthalpy function (cid:12) ( u ). We ﬁrst establish the optimal regularity of weak solutions u ≥ 0 for one phase problem. Furthermore, we show that the free boundary @ { u > 0 } is locally Lipschitz continuous graph provided that v = e N , the direction of x N coordinate axis and @ z u ≥ 0. The latter monotonicity assumption in z variable can be easily obtained for a suitable boundary condition. variable z .


Introduction
In this article we study the optimal regularity of weak solutions to the stationary Stefan problem, with prescribed convection, and the smoothness of free boundary. There are a number of phase transition problems in applied sciences that are encompassed by this mathematical model, among which is the thawing or freezing of the water where the liquid part is in motion, for more details we refer to [4], [1] Chapter 10.7, [11].
In general setting the convection term v is to be determined from a Navier-Stokes system [4], however in this paper we assume that v is given. Furthermore, in the study of regularity of free boundary we will consider constant convection vector v and take f = 0, [11]. The phase transition problems with prescribed convection is called the continuous casting problem, and appears for instance in metal production [11] page 32.
Here we focus on a model anisotropic stationary problem with uniformly elliptic matrix Aij(x) with C 1,α , α > 0 regular entries which are independent of "height" variable z.

Problem set up
We now turn to the mathematical formulation of the problem. Let Ω ⊂ R N −1 be a bounded Lipschitz domain. Let L > 0 and set CL = Ω × (0, L). The points in CL will be denoted by X = (x, z), where x ∈ Ω and z ∈ (0, L). The partial derivatives of a function u : CL → R are denoted by ∂x i u, ∂zu, i = 1, . . . , N −1.
Sometimes we will write ∂iu or ui instead of ∂x i u for short. For more background on this problem see [10]. It follows from (2.2) that u satisfies We will be also interested in the local behaviour of weak solutions of (2.4) div (A(x)∇u) = ∂zβ(u) + f in CL.
Throughout this paper we make the following hypotheses on the matrix A: In other worlds A is independent of z variable, uniformly elliptic with C 1,α 0 continuous entries. Proof. The proof, which we briefly sketch here, is standard and is based on penalisation method [3], [5]: for any ε > 0 we consider the boundary value problem From (2.5) it follows that there is unique u ε ∈ H 1 (Ω) ∩ C 2,α (CL) for some α ≤ α0. Furthermore, if one multiplies this equation by u ε − g then after standard manipulations we can get Hence, choosing δ > 0 small enough and after rearranging the terms we get ) with some tame constant C independent of ε. From here and Poincaré's inequality [6] we get the uniform estimate ∥u ε ∥ H 1 (C L ) ≤ C. After passing to the limit one can readily verify that the limit function u solves the equation LAu = auz + ηz in the weak sense and η takes values only in the interval [0, ℓ]. The Hölder continuity follows from the standard DeGiorgi type estimates.
For reader's convenience I will give the proof of Proposition 2, which is similar that of [3] with slight amendments due to the anisotropy of A in the last section of the paper. Note that (2.6) is necessary for Proposition 2 to hold, see [3].

Corollary 1.
Retain the conditions of Proposition 1 and assume further that there is c0 > 0 such that Then u is monotone in z direction and ∂{u > 0} is C α graph over Ω.
The proof of Corollary 1 follows from Proposition 2 and (2.7) and can be found in [3]. It is worth noting that the method in [3] gives the same degree of regularity for both the solution in CL and the free boundary on Ω. Unfortunately, the best global regularity for u one can expect, under condition of Proposition 2 is log-Lipschitz. On the other hand the best local regularity of u that is Lipschitz continuity, see Theorem 1. However, in local outset the strong monotonicity of u in z−variable does not follow immediately and some delicate analysis is required in order to obtain the strong monotonicity of u in the subdomains of CL.
Now we formulate our main results.

Theorem 1.
Let u be a non-negative bounded weak solution to (2.2). Then u is locally Lipschitz continuous in CL, provided that v ∈ L ∞ (CL, R N ) and f ∈ C(CL).
The local regularity for two phase problem is discussed in [8], and [9]. As for the regularity of free boundary, our main result here states that if u is a Lipschitz continuous solution of (2.2) and ∂zu ≥ 0, then the free boundary is a locally Lipschitz continuous graph in z−direction.
Theorem 2. Let u be a nonnegative weak solution to (2.1) in CL such that u is nondecreasing in z−direction. Then for any subdomain D ⊂ CL, Γ(u) = ∂{u > 0} ∩ D is locally a Lipschitz graph in Before entering into the details of the proof we would like to highlight the main ideas in the proof of Theorem 2. First we establish the non-degeneracy of u. Then it will be seen that ∂zu ≥ 0 implies strong monotonicity ∂zu ≥ c0 > 0, for some c0 = c0(D), locally for any subdomain D ⊂⊂ CL. Combining this with the Lipschitz continuity of u the proof will follow.
The paper is organised as follows: In Section 3 we prove the local optimal regularity of the weak solutions of (2.2). In Section 4 we introduce Baiocchi's transformation w of u which allows us to retrieve the non-degeneracy of u form that of w, which solves an obstacle like problem. The non-degeneracy of u, established in Section 5,, is crucial in our analysis, especially in the proof of strong monotonicity in z−variable, see Proposition 3. The proof of the main regularity result for free boundary is contained in Section 6. Finally, last section contains the proof of comparison principle, Proposition 2.

Optimal Growth
By Proposition 1, u is bounded. Moreover the weak solutions of (2. 3.1. Proof of Theorem 1. As it is pointed out in [7], it is enough to show that for any compact set Assume that this inequality is false. Then there exist a sequence of weak solution uj such that Since the weak solutions uj are bounded it follows from (3.1) that M > j2 −k j implying that kj → ∞.
According to (2.2), vj solves the following equation Similarly we obtain sup ). Moreover, it follows ). Thus

Baiocchi's transformation and its properties
In this section we study the weak solutions u of the continuous casting problems which are monotone in z variable, i.e. ∂zu ≥ 0. The monotonicity in z variable can be achieved for a suitable choice of boundary data [3], see (2.7).
We establish the key estimate for weak solutions of (2.2), which will be used in the proof of Theorem 2. Our first lemma is of technical nature linking u with the solution of obstacle problem via Baiocchi's transformation. Recall that Baiocchi's transformation w of u is defined by From definition it follows that w is convex in z variable provided that ∂zu ≥ 0. Proof. By direct computation we have The first term is´z 0 LAu(x, s)ds = au + ℓχ {u>0} . It remains to combine the second and fourth line in the computation in order to obtain where to get the second line we used ∂x N Aij = ∂zAij = 0. Now the proof is complete.
and data such that Proof. Suppose that (4.2) fails. Then there is a sequence kj such that Using the same reasoning as in Theorem 1 we conclude from (4.3) that the scaled functions wj( has the properties with scaled matrix Aj(X) = A(Xj + rjX), X ∈ B1. From (4.3) we see that Recalling (4.4) and utilizing (i)-(iv) we see that that w0 is a non-negative, non-zero A0-harmonic function in B 3/4 such that w0(0)=0, which however is in contradiction with the maximum principle. The proof is complete.
We close this section by proving the non-degeneracy of w. is always true. Hence we conclude that We want to show that sup Applying the maximum principle to η we get η < 0 in Br(X0) ∩ {w > 0}. From η < 0 we also see that that w(X0) < 0 which is a contradiction.

Non-degeneracy of u
Now we turn to the non-degeneracy of weak solution u to the continuous casting problem. and the data such that for any BR(X0) ⊂ D, X0 ∈ ∂{u > 0} there holds The proof of (5.1) is by contradiction. Suppose that for some fixed D ⊂ CL with dist(D, ∂CL) > 0 there are Rj > 0, Xj ∈ D ∩ ∂{u > 0} such that It follows that wj solves the equation Furthermore, wj has the following properties where C is independent of j and C0 = ℓ 8N Λ , see Lemma 3. Using a standard compactness argument we can extract a subsequence {jm} such that (i) wj m ⇀ w0 weakly in H 1 (B1) for some function w0 ∈ H 1 (B1), We claim that div( A0∇w0) = 0 in B1. Indeed, from (5.2) we infer that χ {u j >0} → 0 almost everywhere.
Next, we record some properties of the blow up limits. Recall that the blow up limit of u at X0 is defined as v0(X) = lim

Proof. It is enough to notice that sup
Bs u(X 0 +r j X) r j ≥ C1 for a fixed s > 0 and small rj. To see this one needs to apply Lemma 4 and use a customary compactness argument.

Corollary 2. Let v0 be as in Lemma 5, then there is a constant CD such that
Proof. If not then there exist Xj ∈ ∂{v0 > 0}∩D and a sequence 0 < rj ↓ 0 such that ffl Since ∇vj(X) = ∇v0(Xj + rjX) and v0 is Lipschitz, it follows from Arzelà-Ascoli theorem that there exists a subsequence j k such that vj k (X) → V (X) uniformly in B1 for some function V . In particular ffl B 1 V 2 = 0. However this contradicts the non-degeneracy of v0, Lemma 4, because

Corollary 3. Let v0 be as in Lemma 4. Then there exists C ′ D > 0 such that
Proof. We argue as in the proof of the previous Corollary. Thus there are Xj ∈ ∂{v0 > 0} ∩ D and because ∇vj(X) = (∇v0)(Xj + rjX), thus in particular the sequence {vj} is uniformly Lipschitz continuous in B1. By a customary compactness argument we can extract a subsequence j k such that vj k → V uniformly in B1 and ∇vj k ⇀ ∇V weakly in B1.
By the semicontinuity of Dirichlet integral we get implying that V ≡ 0 in B1 (recall that v0(Xj) = 0 which translates to vj k (0) = 0). But this contradicts the non-degeneracy of v0, see (5.4), because 1 r j sup Br j (X j ) v0 = sup B 1 vj ≥ cD and by uniform convergence this yields sup B 1 V ≥ cD.
We close this section by giving an application of Corollary 3, see [2]. It provides a rough estimate for the measure of a neighbourhood of free boundary and will be used in the proof of strong monotonicity of u in the next section. Lemma 6. Let v0 be as in Lemma 5. Then there is a tame constant C > 0 such that for any R, and small σ, 0 < σ < R the following inequality holds Notice that t < vσ,t ≤ σ and v0 is Lipschitz continuous, thereby Sending t to zero we conclude Next, we define the maximal distance of {v0 = σ} from ∂{v0 > 0}, i.e. d = sup By Lipschitz continuity, Theorem 1, v0(X) ≤ 4Cσ∥Dv0∥∞ for any X ∈ B4Cσ(∂{v0 > 0}) ∩ {v0 > 0}. Thereby Combining (5.6), (5.7) and (5.5) we get and we arrive at the desired inequality.

Lipschitz regularity of free boundary
Now we are ready to demonstrate the strong monotonicity of u in the z−direction.

Proposition 3. Let u be the weak solution to (2.1) such that (5.4) holds.
Then there exist c1 > 0 such that we have Proof. The proof is by contradiction. Suppose that (6.1) fails, then there are points Xj ∈ D ∩ Γ such that lim inf Y →X j ∂zu(Y ) < 1 j and there exists Yj ∈ {u(X) > 0} such that Let Xj ∈ ∂{u > 0} be such that the distance ρj def ≡ dist(Yj, ∂{u > 0}) is realized and ρj = | Xj − Yj|.
Thus for any ε > 0 there is δ > 0 such that uniformly in j Since Yj ∈ B2 and by Arzelà-Ascoli theorem there is a subsequence j k and a function v0 ∈ C 2 (B2) such that where A0 is some constant positive definite matrix (thanks to condition (2.5)), H is the Heaviside function and the last inequality follows from (5.4), the definition of vj and the uniform convergence of vj k in B2.

Proof of Theorem 2. From Proposition 3 and Theorem 1 we have
Let h(x) = inf{z, u(z, x) > 0} the height function of the free boundary over x ∈ Ω. Thanks to ∂zu ≥ c1 > 0, h is continuous and the free boundary is a continuous graph over Ω. Then for small ε > 0 we take z2 = h(x1) Swapping x1 and x2 and letting ε → 0 the result follows.

Proof of Proposition 2
The proof is very similar to [3] Lemma 2.1, however there are technical complications due to the heat condition coefficients Aij.
Using ξ ∈ C ∞ 0 (CL), ξ ≥ 0 in the weak formulation of solution u and supersolution u ⋆ we get After integration by parts we get Since v ≥ 0 and v(X0) = 0 it follows Notice that X0−Aνt ∈ CL if t > 0 is small enough thanks to the ellipticity of A. Thus lim (∇ξAν)(X0) and the claim is proved.
As for the remaining two integrals we first set notice that And again we see that we only need to estimate the normal derivative of ξ n on Ω × {L}.
We first prove uniform C 0 bound for ξ n in order to estimate I1 and then an estimate for ∂ν ξ n on Ω × {L}.
Taking ξ = ξ n in (7.2) we get implying η ⋆ ≥ η in CL, and the proof is complete.