REGULARITY RESULTS FOR A QUASILINEAR FREE BOUNDARY PROBLEM

In this paper we prove local interior and boundary Lipschitz continuity of the solutions 
 of a quasilinear free boundary problem. We also show that the free boundary is the union of graphs of lower 
 semi-continuous functions.


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(ii) For the existence of a solution of problem (1.1), one usually needs to impose some boundary conditions on ∂Ω, which are typically a mixture of Dirichlet and Neuman conditions. Then under assumptions (1.2) and (1.4), one can prove existence of a solution to problem (1.1) by arguing as in [16,17,20,21,23]. The main idea consists in solving an approximating problem using Schauder fixed point theorem, and then passing to the limit using adequate estimates.
Among free boundary problems that fit in the problem (1.1) setting, is the dam problem (see [4,5,8,12,20,21,22,23], which consists in studying the filtration of a fluid through a porous medium Ω ⊂ R n , where we look for the fluid pressure or hydrostatic head pressure inside Ω and the saturated region represented by a function that lies between 0 and 1. The classical formulation of the dam problem assumes that the flow is governed by Darcy's law i.e. where u is the fluid pressure, e = (0, ...0, 1), v is the fluid velocity and a(x) is the matrix permeability of the porous medium. In this case equation (1.1)ii) reads (see [17,20]) div (a(x)(∇u + χe)) = 0 in H −1 (Ω).
It is well known that Darcy's law fails to hold for non-Newtonian fluids in which case it is substituted by a power-law of the form v = −k|∇u| p−2 ∇u, where u is the fluid hydrostatic head pressure=fluid pressure+x n , v is the fluid velocity and k is a positive constant. If we set g = 1 − χ, then equation (1.1)ii) reads (see [4]) In order to take into account the heterogeneity of the medium and the non-Newtonian flow, the following generalization of the above power-law was proposed in [21] (see also [5,12,22,23]): v = A(x, ∇u), where A is a vector function from Ω×R n to R n such that A(., ξ) is measurable, A(x, .) is continuous and monotone, A(x, ξ).ξ ≥ λ|ξ| p and |A(x, ξ)| ≤ Λ|ξ| p−1 for a.e. x ∈ Ω and all ξ ∈ R n , for some p > 1 and λ, Λ > 0. In this case, (1.1)ii) reads (see [5,12,21,22,23]) div (A(x, ∇u) − gA(x, e)) = 0 in W −1,p (Ω).
Another application of problem (1.1) arises from the lubrication problem (see [1,2]) which describes lubrication with cavitation in bearings. The classical formulation of this problem assumes that the flow in a rectangular domain Ω is governed by Reynolds law i.e. div h 3 (x 1 )∇u = h (x 1 ) in {u > 0}, where u is the fluid pressure, and h(x 1 ) is the gap between the bearing and the shaft. In this problem there are two unknowns, the fluid pressure u and the fluid relative thickness 0 ≤ χ ≤ 1. If e 1 = (1, 0) then Reynolds law and the incompressibility of the fluid lead to the following version of (1.1)ii) (see [1,2] One more application of problem (1.1) is the thermoelectrical modeling of aluminium electrolysis (see [3]). This model is based on the Fourier law q = −k(x)∇T , where T is the aluminium temperature in an electrolytic cell section materialized by a bounded domain Ω of R 2 , q is the heat flux and k(x) is the thermal conductivity. Assuming that T s is the solidification temperature of aluminium, then the problem consists in finding the function u = T − T s ≥ 0 and a function 0 ≤ χ ≤ 1 that describes the region occupied by the liquid phase {u > 0}. If h(x) represents the heat flux through the free boundary ∂{u > 0} ∩ Ω, then the Fourier law and the conservation of energy equation lead to the following version of (1.1)ii) (see [3]) For a more general framework, we refer to [6,7,9,11,15,24]. In this paper we generalize results from [9,11] for the p-Laplacian and results in [6,7] in the linear case. Regarding the problem with a Newman boundary condition, we refer to [17] for the dam problem, and to [25,27] for a more general framework.
In the first part of the paper, we show interior and boundary Lipschitz continuity. In the second part, under more assumptions on H including div(H) ≥ 0, we establish that the free boundary is represented by a family of lower semicontinuous functions.
Throughout this paper, we shall denote by B r (x) an open ball with center x and radius r. If the center is not given, it will be assumed to be the origin.

Interior and boundary Lipschitz continuity
The first result of this section is the following interior regularity. [19]. Therefore to prove Theorem 1, it is enough to investigate the behavior of u near the free boundary. This is the object of the following lemma.
Proof. We start by applying Harnack's inequality (see [19], Corollary 1.4): where C is a positive constant depending only on n, a 0 and a 1 . Therefore, to prove the lemma, it will be enough to establish the inequality and we can define the following function in the circular ring We claim that Indeed, we first observe that Taking into account the fact that we get by substituting the above formulas in (2.2) Hence (2.1) holds, which leads by using (1.5) to ≤ h(r + ), and by letting → 0, we get a κ r . me −κ e − κ 4 − e −κ ≤ hr ≤ hδ(Ω), which leads to m ≤ a −1 hδ(Ω) (e 3 4 κ − 1) κ r = C(n, a, h, δ(Ω))r and the lemma follows.
Proof. (of Theorem 1) The proof is based on Lemma 1 and arguments similar to those in the p-Laplacian case [11]. In particular, we use the scaling function v(y) = u(x 0 + Ry)/R for y ∈ B 1 , which satisfies the equation Then by applying the estimate from [19], Theorem 1.7, we get for some positive constant C(n, a, M, R) that sup B 1/2 |∇v| ≤ C(n, a, M, R).
Now we assume that u = 0 on a nonempty subset T of ∂Ω, and we study the Lipschitz continuity of u up to T . To this end we assume the uniform exterior sphere condition satisfied locally on T i.e. for each open and connected subset S ⊂⊂ T ∃R > 0 such that ∀y ∈ S ∃z ∈ R n \ Ω B R (z) ∩ S = {y}.
Without loss of generality, we can assume that R < 1 3 dist(S, ∂Ω \ T ) > 0, where dist is the distance between two sets. Then we state our second result.
The proof of Theorem 2 is based on Lemma 2. The rest of the proof will be omitted, since it can be easily obtained using arguments similar to those in the proof of Theorem 1 [11] and taking into account the above remark at the end of the proof of Theorem 1. Then it is easy to verify that the following properties of ϑ hold: We also have (d(x))).
Therefore, since |x − z| > R for all x in Ω, we obtain Next, we claim that . Using this function in (1.1)ii) and in (2.7), we obtain Taking into account that χ = 1 a.e. in {u > 0} and adding (2.9) and (2.10), we obtain which leads by (1.3) to ∇(u − v) + = 0 a.e. in Ω, and therefore (u − v) + is constant in Ω. Since u ≤ v on ∂Ω, we get u ≤ v in Ω. We conclude that for all x ∈ Ω and y ∈ S, we have

The free boundary
In this section, we assume that the vector function H satisfies the following assumptions for some positive constants h and h: By using min u/ , 1 ζ with ζ ∈ D(Ω), ζ ≥ 0 as a test function for (1.1)ii) and arguing as in [7], one can establish the following important inequality: As a consequence of (3.3), we will derive a weak monotonicity of the function χ, that will be used to express the free boundary as a union of graphs of a family of functions. More precisely, we consider the following differential equation where h ∈ π xn (Ω) and ω ∈ π x (Ω ∩ {x n = h}), x = (x 1 , ..., x n−1 ), π x and π xn are respectively the orthogonal projections on the hyperplane {x n = 0} and the x n -axis. Then we denote by X(., ω, h) the maximal solution of E(ω, h) defined on the interval (α − (ω, h), α + (ω, h)). We deduce from (1.4) that we have It follows that the limits lim t→α−(ω,h) + X(t, ω, h) and lim t→α+(ω,h) − X(t, ω, h) both exit, which we shall denote respectively by X(α − (ω, h), ω, h) and X(α + (ω, h), ω, h), and observe that we have necessarily X(α − (ω, h), ω, h) ∈ ∂Ω ∩ {x n < h} and For simplicity, we will drop the dependence on h in the sequel. Now, we recall for the reader's convenience the following technical properties and definitions established in [7]: • α + and α − are uniformly bounded.
• For each h ∈ π xn (Ω), we define the set D h = {(t, ω) / ω ∈ π x (Ω ∩ {x n = h}), t ∈ (α − (ω), α + (ω))} and consider the mapping • The determinant Y h (t, ω) of the Jacobian matrix of the mapping T h , satisfies: Using (3.3) and arguing as in the proof of Theorem 1 of [7], we can establish the following monotonicity of χ Property (3.4) means that χ decreases along the orbits of the differential equation (E(w, h)). The consequence on u is materialized in the next key theorem which is the main idea in the parametrization of the free boundary.
Theorem 3. Let (u, χ) be a solution of (1.1) and To prove Theorem 3, we need the following strong maximum principle.
The proof of Lemma 3 follows from the next Lemma 4 as in [18] p. 333.

Let = min
To establish the Lemma, we will compare u − u(x 0 ) with respect to v. We claim that Indeed, we first observe from (2.2) that (3.7) Moreover, we have Taking into account the fact that we get by substituting the above formulas in (3.7) Hence (3.6) holds. Using (3.6) and the fact that ∆ A u ≤ 0, we obtain To conclude, let ν be the exterior unit normal vector to ∂B R (x 1 ) at x 0 . We infer from (3.9) for t positive and small enough so that Letting t −→ 0, we obtain Proof. (of Theorem 3) It is enough to verify i). By continuity, there exists > 0 such that uoT h (t, ω) > 0 ∀(t, ω) ∈ (t 0 − , t 0 + ) × B (ω 0 ) = Q . By (1.1)i), we have χoT h (t, ω) = 1 for a.e. (t, ω) ∈ Q . Using (3.4) and the fact that χoT h ≤ 1, we get χoT h = 1 a.e. in C , i.e. χ = 1 a.e. in T h (C ).
From (1.1)ii) and (3.2), we get A u = −div(H) ≤ 0 in D (T h (C )). Given that u ≥ 0 in Ω and u > 0 in T h (Q ) ⊂ T h (C ), we conclude by Lemma 3, that u > 0 in T h (C ).