Lipschitz Continuity for Elliptic Free Boundary Problems with Dini Mean Oscillation Coefficients

We establish local interior Lipschitz continuity of the solutions of a class of free boundary elliptic problems assuming the coefficients of the equation of Dini mean oscillation in at least one direction. The novelty in this regularity result lies in the fact that it allows discontinuous coefficients in all but one variable.


Introduction
Throughout this paper, we denote by Ω a bounded domain in R n and by A(x) = (a ij (x)) an n × n matrix that satisfies for some positive constant λ ∈ (0, 1) i,j |a ij (x)| ≤ λ −1 , for a.e.x ∈ Ω, ( A(x) • ξ • ξ ≥ λ|ξ| 2 , for a.e.x ∈ Ω, for all ξ ∈ R n (1.2) f : Ω → R n is a vector function such that f (x) = (f 1 (x), ..., f n (x)) and We consider the following problem (P ) This class covers a set of various problems including the heterogeneous dam problem [1] [4], [8], [15], in which case Ω represents a porous medium with permeability matrix A(x), and f (x) = A(x)e, with e = (0, ..., 0, 1).A second example is the lubrication problem [2] which is obtained when A(x) = h 3 (x)I 2 and f (x) = h(x)e, where I 2 is the 2 × 2 identity matrix, and h(x) is a scalar function related to the Reynolds equation.A third example is the aluminium electrolysis problem [3] which corresponds to A(x) = k(x)I 2 and f (x) = h(x)e, with k(x) and h(x) two given scalar functions.
We observe that if f ∈ L q loc (Ω) for some q > n, then so is χf , and by taking into account the assumptions (1.1)-(1.2) and the equation (P )ii), we infer from [14] Theorem 8.24, p. 202 that u ∈ C 0,α loc (Ω) for any α ∈ (0, 1).In this paper, we will improve this regularity by showing that under suitable assumptions, we actually have u ∈ C 0,1 loc (Ω).We observe that this regularity is optimal due to the gradient discontinuity across the free boundary which is the interface that separates the sets {u = 0} and {u > 0} from each other.Moreover, Lipschitz continuity is not only interesting by itself, but is also of particular importance in the analysis of the free boundary (see for example [7] and [9]).
Before stating our main result, we need to introduce a definition.
satisfies the Dini condition.iii) For each f ∈ L 1 (Ω), we define the following functions: , then it is easy to verify that ω f (r) ≤ 2Cr α for any r ∈ (0, 1], which leads to Hence, f is of partial Dini mean oscillation with respect to x ′ in any open ball B ⊂⊂ Ω.
Here is the main result of this paper: Theorem 1.1.Assume that A and f satisfy (1.1)-(1.3)and the following conditions: ∀i, j = 1, ..., n, a ij is of partial Dini mean oscillation with respect to x ′ in Ω (1.4) ∀i = 1, ..., n, f i is of partial Dini mean oscillation with respect to x ′ in Ω (1.5) Then for any weak solution of (P ), we have u ∈ C 0,1 loc (Ω).
The novelty in Theorem 1.1 lies in the fact that Lipschitz continuity of weak solutions of problem (P ) is obtained even when the entries of the matrix A(x) and the vector function f (x) are discontinuous provided they satisfy a Dini mean oscillation condition in at least one direction i.e. if they are regular in at least one variable.Since problem (P ) is invariant by rotation in the sense that it is transformed into a similar problem with different coefficients satisfying the same assumptions as the original ones, it is obvious that we only need to have the Dini mean oscillation condition satisfied in any arbitrary space direction.
We recall that interior Lipschitz continuity for problem (P ) was established in [7] and the same method was successfully extended to the quasilinear case in [10] and [11].Interior and boundary Lipschitz continuity were established in [16] for a wide class of linear elliptic equations under some general assumptions.Recently, in [17], Lipschitz continuity was obtained using a different method based on Harnack's inequality.This approach helped relax some of the assumptions required in [7] and [16] and only required that A(x) ∈ C 0,α loc (Ω) and div(f ) ∈ L p loc (Ω) for some α ∈ (0, 1) and p > n/(1 − α).
Lastly, we would like to point out that the assumptions (1.4)-(1.5)were introduced in [13] to obtain C 1 and C 2 -regularity of solutions to elliptic equations.In this regard, we also refer the reader to the recent work on gradient estimates for elliptic equations in divergence form with partial Dini mean oscillation coefficients [6] .
2 Estimates for the equation div(A(x)∇u) = −div(f ) The main result of this section is a local L ∞ −norm estimate of the gradient which will be used in the proof of Theorem 1.1 in section 3. Needless to say, this estimate is of interest for itself.
) and A and f are of partial Dini mean oscillation with respect to x ′ in B 2ρ (x 0 ), then ∇u ∈ L ∞ (B ρ (x 0 )) and we have for some positive constant C 1 depending only on n and λ: The proof of Theorem 2.1 requires a few lemmas.
Lemma 2.1.Assume that ω is a Dini function and let a ∈ (0, 1) and b > 1 be two given real numbers.Then the function defined by In particular, ω is also a Dini function.
Proof.First, we recall that the function ω was introduced in [[12], Lemma 3.1], where an estimate was also given.Nevertheless, our estimate is new and more precise.We start by writing ω(t) = ω 1 (t) + ω 2 (t) for t ∈ (0, 1), where and observe that we have t ≤ b −i iff i ≤ i 0 .Then we have Given that i 0 ≤ − ln(t) ln(b) < i 0 + 1, we can write Moreover, since a < 1, we have Now, combining (2.1) and (2.2), we obtain The following lemma is a slight improvement of Theorem 1.2 of [6] in the sense that it provides a more precise L ∞ -estimate of the gradient. where ) is a positive constant depending only on n and λ, and k 0 is an integer greater than 1 satisfying Proof.First, we observe that by scaling, we may replace the ball B 3 by B 6 as in [6]. .Let now k 0 be a positive integer greater than 1 that satisfies which by taking into account the estimate of Lemma 2.1 is true if At this step, we further assume that κ ≤ 2 −2 , which leads to 1 − √ κ ≥ 2 −1 , and makes the above inequality hold if This in turn remains true if κ and k 0 are chosen such that If we replace f 1 by f n , we get the estimate [see [6], p. 1522] with a positive constant C 1 (n, λ) depending only on n and λ which can be written by using Lemma 2.1 again as The following lemma is a slight improvement of Lemma 2.2 of [17].
Lemma 2.3.Assume that u is a nonnegative weak solution of the equation div(A(x)∇u) = −div(f ) in Ω and let x 0 ∈ Ω and r > 0 such that B 5r (x 0 ) ⊂⊂ Ω and B r (x 0 ) ∩ {u = 0} = ∅.Then we have for some positive constant C 2 depending only on n, λ and f : max Given that B r (x 0 ) ⊂ B 2r (x 1 ), the lemma follows.
The following lemma is a Cacciopoli type lemma.
Lemma 2.4.Assume that u is a weak solution of the equation div For each open ball B r (x 0 ) such that B 2r (x 0 ) ⊂ Ω, we have: Proof.Let B r (x 0 ) be an open ball such that B 2r (x 0 ) ⊂ Ω, and let η ∈ C ∞ 0 (B 2r (x 0 )) be a cut-off function such that Using η 2 u as a test function for equation (P )ii), we get which can be written as Combining Lemmas 2.3 and 2.4, we obtain the following lemma.

Lemma 2 . 2 .
Let u ∈ H 1 (B 3 ) be a weak solution of the equation div(A(x)∇u) = −div(f ) in B 3 , with A and f satisfying (1.1)-(1.3) in B 3 and both A and f of partial Dini mean oscillation with respect to x ′ in B 2 .Then we have ∇u ∈ L ∞ (B 1 ) with Next, we denote by C 0 = C 0 (n, λ) the positive constant depending only on n and λ that was introduced in [[6], Proof of Theorem 1.2, p. 1520].Following this reference, we choose γ = 1 2 , 0 < κ < min(2 −1 , C −2 0 ) and we denote by ω A (t) the function defined in Lemma 2.1 with a = √ κ and b = 1 κ