Regularity for the two phase singular perturbation problems

We prove that an a priori BMO gradient estimate for the two phase singular perturbation problem implies Lipschitz regularity for the limits. This problem arises in the mathematical theory of combustion where the reaction-diffusion is modelled by the $p$-Laplacian. A key tool in our approach is the weak energy identity. Our method proves a natural and intrinsic characterization of the free boundary points and can be applied to more general classes of solutions.


Introduction
A chief difficulty in working with two phase nonlinear free boundary problems is the absence of monotonicity formulas. One question still unanswered is whether the weak or variational solutions to the two phase free boundary problems have optimal regularity. These solutions, for instance, arises in the mathematical theory of combustion, in the models of high activation of energy.
In this paper we address this question by developing a unified approach for the nonlinear two phase free boundary problems. To elucidate our main ideas we start from the singular perturbation problem for the p-Laplacian.
Let u ε be a family of solutions to where ε > 0 is a parameter and The quasilinear operator ∆ p u ε = div(|∇u ε | p−2 ∇u ε ), 1 < p < ∞ is called the p-Laplacian. If p > 2 then ∆ p is degenerate elliptic, and it depicts diffusion obeying power law.
1.1. Known results. Zel'dovich and Frank-Kamenetskii studied the one dimensional version of (P ε ) in 1938, see [ZBLM85] Chapter 1.4, and calculated the speed of the front which is √ 2M, see formula (4.30) there. In high dimensions the one phase problem for the laplacian was studied by Berestycki, Caffarelli and Nirenberg [BCN90], Caffarelli Vazquez [CV95], and the two phase problem in [Caf95], [CLW97] for the heat equation ∆u ε − ∂u ε ∂t = β ε (u ε ). Later Caffarelli and Kenig studied the two phase problem for the variable coefficient case div(a∇u ε ) − ∂u ε ∂t = β ε (u ε ) [CK98]. The key tool the authors used in the proof of optimal regularity is the monotonicity formula of Caffarelli [Caf93]. With its help one can establish local uniform Lipschitz estimates in the parabolic distance.
1.2. Heuristic discussion. Heuristically, the limits of u ε as ε → 0 are the solutions of the two phase Bernoulli type free boundary problem In order to pass to the limit we need some uniform continuity for u ε , say u ε L ∞ (B 1 ) ≤ 1. Then from Caccioppoli's inequality it follows that sup ε u ε W 1,p (B R ) ≤ C(R, n, p), for every R < 1. Assume that p > n, then from Sobolev's embedding theorem we infer that u ε is locally uniformly Hölder continuous. Consequently, if u(x) > 0 for some x ∈ B 1 , then u is p-harmonic in some neighborhood of x. Moreover, the uniform convergence also implies that ∇u ε j → ∇u strongly in L p loc (B 1 ).
Using these observations it is easy to see that the limits u of singular perturbation problem satisfy the weak energy identity |∇u| p + pB * (x)) divX = p |∇u| p−2 ∇u∇X∇u, where X is a C 1 vectorfield with support in B 1 . The function B * is bounded and characterizes the concentration of the measure ∆ p u ε on the free boundary ∂{u > 0} as ε → 0. If ∂{u > 0} is C 1 then one can see that B * = Mχ {u>0} with M = β, see Lemma 7.1. In particular, every stationary point of the functional B 1 |∇u| p + Mχ {u>0} satisfies the weak energy identity.
We remark that there may be solutions u, obtained as limits of (P ε ), of the form x 0 with α,ᾱ ≥ 0 and at these points the stratification argument [DK18] fails. This is the reason why we further split the flat points into to parts and use the Lebesgue density to identify the points where u is a viscosity solution.
We claim that (1.3) implies that ∆ p u 0 = 0 in R n . The converse statement is obviously true since (1.3) is the domain variation of the energy |∇u| p . This is an interesting question of independent interest since (1.3), as we show in this paper, gives another characterization of the p-harmonic functions for p > n. That done, we can apply Liouville's theorem to conclude that u 0 is a linear function. Combining this with the stratification argument from [DK18] with respect to the modulus of continuity of the slab flatness and the Lebesgue density of {u < 0} we conclude that if either (A) or (B) hold then u has linear growth at x 0 . For the remaining case (C) we can conclude that u is a viscosity solution and x 0 is flat, so from the Harnack principle we infer that ∂{u > 0} near x 0 is C 1,γ smooth hypersurface, and the linear growth for this case follows from the standard boundary gradient estimates for u.
Suppose x 0 is a free boundary point Theorem 1.1. Let u ε j be a family of solutions to (P ε ) such that u ε j → u in W 1,p then u is Locally Lipschitz continuous.
The paper is organized as follows: Section 2 contains some well known estimates for the solutions and subsolutions of the p-Laplacian.
In Section 3 we prove the weak energy identity. If the Lipschitz continuity fails then we get that the weak energy identity simplifies. In Section 4 we show that BMO solutions satisfying this simplified identity must be p-harmonic Then we consider the scenarios (A), (B) and (C) in Sections 5 (Flatness vs linear growth), 6 (density of negative set) and 7 (viscosity solutions), respectively.
Combining our results in Section 8 we give the proof of Theorem 1.1 In Section 9 we study the properties of solutions with Lipschitz regularity. Section 10 is devoted to the weak solutions. Finally, in Section 11 we prove a BMO type estimate for the tensor T ij .

Notation
We fix some notation. The n dimensional Euclidean space is denoted by R n , u + = max(u, 0) is the nonnegative part of u and similarly u − = − min(u, 0), so that u = u + − u − . The partial derivatives in x i , i = 1, . . . , n variable are denoted by ∂ i u or u i so that ∂ i u = ∂u ∂x i , (x − x 0 ) 1 is the first slot function of the vector x − x 0 . For every u ∈ W 1,p (B 1 ) we also define the tensor T lm (∇u) = p|∇u| p−2 u l u m − |∇u| p δ lm . Sometimes we let Γ = Γ u to denote the free boundary ∂{u > 0} when no confusion can arise, Vol(E) denotes the n-dimensional volume of a set E.

Preliminaries and tools: Uniform estimates and compactness
We recall the well known inequality [DM93] (2.1) 2.1. Caccioppoli inequality and local compactness.
Lemma 2.2. Let u ε be a family of solutions of (P ε ), then exists a constant C = C(n, p) > 0 depending only on n, p such that for every R ∈ (0, 1) there holds Proof. Let η ∈ C ∞ 0 (B 1 ), 0 ≤ η ≤ 1, |∇η| ≤ C/R for some C = C(n) > 0, and η = 1 on B R . From (1.1) it follows that u ε β ε (u ε ) ≥ 0. Thus using u ε η p as a test function in the weak formulation of the equation Rearranging the terms and using the Hölder inequality we obtain From here we see that and the desired estimate follows with C(n, p) = p p C p (n).
In the next Proposition we assume p > n, then from (2.2), the assumption |u ε | ≤ 1, and Sobolev's embedding theorem it follows that for fixed 0 < R < 1 and uniformly in ε. Proposition 2.3. Let u ε be a family of solutions to (P ε ) and p > n. Then the following statements hold true: for every sequence ε k → 0 there is a subsequence, still labelled ε k , and a function u such that Proof. From Lemma 2.2 and Sobolev's embedding theorem it follows that Thus (i) follows from a standard compactness argument.
The proof of (ii) is standard, see [DM93].
To prove (iii) it is enough to show that This and the weak convergence imply strong convergence. It follows from (1.1) that u ε β ε (u ε ) ≥ 0. Using u ε ψ as test function in the weak formulation of the equation we get as in the proof of Lemma 2.2 Applying Fatou's lemma we obtain Let s > 0 be a small number. Then (u − s) + ψ ∈ W 1,p 0 ({u > 0}). Therefore After sending s → 0 we conclude This and (2.9) imply Consequently, Remark 2.4. The assumption p > n is technical. It allows to get the uniform continuity of u ε , see also the discussion in Section 11.
Lemma 2.5. There exists a constant C = C(n, p) depending only on n, p such that for every B 2R (x) ⋐ B 1 the measure µ = ∆ p u ε satisfies the inequality Proof. Using the divergence theorem we get the estimate Integrating both sides of (2.6) over r ∈ [0, R] we obtain On the other hand dµ.
Combining this with (2.7) we get and (2.5) follows.
Proposition 2.6. Let u j be a sequence of solutions to ∆ p u j = µ j in B 1 such that sup j u j W 1,p (B 1 ) < ∞ and µ j are Radon measures in B 1 such that suppµ j ⊂ ∂{u j > 0}. If u 0 is a limit of u j then ∇u j → ∇u 0 strongly in L p loc (B 1 ). Proof. Let 0 ≤ ψ ∈ C 1 0 (B 1 ) then u j ψ∆ p u j ≥ 0 in B 1 . Thus we have Let s > 0 be a small number. Then (u 0 − s) + ψ ∈ W 1,p 0 ({u 0 > 0}). Therefore After sending s → 0 we conclude This and (2.9) imply Consequently, We summarize the previous results replacing B 1 by a general domain D.
Proposition 2.7. Let u ε be a family of solutions to (P ε ) in a domain D ⊂ R n , and p > n. Let us assume that u ε L ∞ (D) ≤ A for some constant A > 0 independent of ε. For every ε k → 0 there exists a subsequence ε k ′ → 0 and u ∈ C 1− n p loc (D), such that (i) u ε k ′ → u uniformly on compact subsets of D, 2.2. First and second blow-up. Using Proposition 2.7 we can extract a sequence u ε j for some sequence ε j such that u ε j → u uniformly in B 1 2 . Let 0 < ρ j ↓ 0, x j ∈ ∂{u > 0} and set , where m j are some positive numbers such that sup j ρ j /m j < ∞. Suppose u j is uniformly bounded and u j → U locally uniformly in R n for some function U defined on R n .
The function U is called a blow-up limit of u with respect to free boundary points x j and, in general, it depends on {ρ j } and x j .
The two propositions to follow establish an important property of the blow-up limits, namely that the first and second blow-ups of u can be obtained from (P ε ) for a suitable choice of parameter ε.
Proposition 2.8. Let u ε j be a family of solutions to (P ε ) in a domain D ⊂ R n such that u ε j → u uniformly on D and ε j → 0.
Then there exists j(k) → ∞ such that for every j ≥ j(k) there holds that ε j /ρ k → 0 and Proof. Our proof closely follows that of Lemma 3.2 in [CLW97] where the case of ρ k = m k . We estimate the difference Fix r > 0, then for every δ > 0 and R < r Consequently, we have that Observe that j(k) can be chosen so large that ε j(k) /m k < 1 k . Hence recalling (2.10) and applying Proposition 2.7 we get parts (i) and (ii).
As for (iii) we use the estimate above to get Therefore and (iii) follows.
Finally, recall that the result of previous proposition extends to the second blow-up.
Proposition 2.9. Let u ε j be a solution to (P ε ) in a domain D j ⊂ D j+1 and ∪ j D j = R N such that u ε j → U uniformly on compact sets of R N and ε j → 0. Let us assume that for some choice of positive numbers d n and points x n ∈ ∂{U > 0}, the sequence converges uniformly on compact sets of R N to a function U 0 . Let Then there exists j(n) → ∞ such that for every j n ≥ j(n), there holds ε j n /d n → 0 and 3. Weak energy identity for the solutions of (P ε ) and the first domain variation This section contains the crucial tool for the proof of our main regularity theorem, the weak energy identity which we state below. In what follows we set On the other hand Next we prove that (3.1) is preserved in the limit as ε → 0.
Proof. We have B(u ε /ε) → B * (x) * -weakly in L ∞ loc . By strong convergence of gradients, Proposition 2.3 (iii), Let us show that the functions |∇u ε j | p−2 ∇u ε j ∇X∇u ε j are equiintegrable. Given σ > 0, then there is j 0 and δ > 0 such that Choose j 0 so large that E |∇u ε | p − E |∇u| p < σ 2 ∇X ∞ (which is possible thanks to Proposition 2.3 (iii)) and then by the absolute continuity of the integral of |∇u| p we can choose δ so small that E |∇u| p < σ 2 . Hence the desired result follows.

Domain variation formula for minimizers.
Let λ > 0 be a constant. We show that the local minimizers of where, for the sake of simplicity, we set λ(u) = λ p χ {u>0} . The identity (3.2) is weaker than (3.3) since we do not know the explicit form of B * . Proof Here we used the inverse mapping theorem for φ t : x → y, in particular a well-known identity .
One can easily verify that with I = {δ ij } being the identity matrix. Hence . and, moreover, This completes the proof of (3.3).
It is convenient to introduce the variational solutions of the free boundary problem Remark 3.5. By inspecting the proof of Theorem 8.3 one can see that the local Lipschitz estimate is valid for the variational solutions in B 1 provided that p > n, see Section (??). Also observe that every stationary point of J p is a variational solution as the above computation shows.

Viscosity solutions
Consequently, Thus if u 0 is a blow-up u at y 0 then from Lamma 4.3 [DK18] (see also Lemma 7.4) either u 0 (x) = α|x 1 |, for some α ≥ 0 in a suitable coordinate system or u 0 is linear. Thus to show that u is a viscosity solution at y 0 it is enough to conclude that α = 0.
We can use the construction in [CLW97] of x 1 symmetric solution and assume that u ε j is x 1 symmetric. Hence, thanks to Propositions 2.8 and 2.9, from now on we assume that u ε j is a family of solutions such that u ε j converges to α|x 1 | is some suitable coordinate system. We claim that α = 0. To see this we observe that Let us denote From the divergence theorem and ∂ 1 u ε j = 0 on x 1 = 0, we get On the other hand and using the co-area formula we conclude that Let x 0 ∈ ∂{u > 0} and be the slab of height 2h in unit direction ν. Let h min (x 0 , r, ν) be the the distance of two parallel planes with unit direction ν containing the free boundary ∂{u > 0} in B r (x 0 ), i.e.
Finally, let Note that h(x 0 , r) is non-decreasing in r. We call h(x 0 , r)/2 the slab flatness constant at scale r > 0.

Optimal growth.
Proposition 5.1. Let u ε j be a family of solutions to (P ε ), such that u ε j → u locally uniformly in |u|.
If h 0 > 0 is fixed and h x 0 , 1 2 k ≥ h 0 2 k+1 for some k, then for some positive constant L, that is independent of x 0 and k.
It follows from (5.6) that The basic idea of the proof is to show that the scaled functions converge to a linear function in R n , which will be in contradiction with (5.5). The proof falls naturally into two parts: first establish some uniform estimates for the sequence {v j }, and then prove that the limit is a p-harmonic function in R n . By construction, (5.10) sup Furthermore, from (5.6) we have that which in turn implies that Finally, since u j (x j ) = 0, we have that Next, from (5.9) we get This gives Consequently, letting B * j (x) := B * (x j + 2 −k j x) and substituting ∇v j into (3.2) we get the differential relation (5.13) ∂ l (T lm (∇v j )) = ∂ m (σ p j pB * j (x)).
Note that σ p j B * j (x) → 0 since B * is bounded. Hence, from Propositions 2.3, 2.8 (with m j = S(k j + 1, u j )) and 2.6 we obtain that for any 0 < R < 2 k j there exists a constant C = C(R, p) > 0 independent of j such that with α = 1 − n p . Therefore, by a standard compactness argument, we have that, up to a subsequence, v j converges to some function v as j → +∞ in C α (B R ), and strongly in W 1,p (B R ) , for any fixed R. (5.14) The properties (5.10), (5.11) and (5.12) translate to v so that Thanks to Theorem 4.1, (5.13) and the strong convergence ∇v j → ∇v we conclude that Hence, from Liouville's Theorem we deduce that v must be a linear function in R N . After rotation the coordinate system we can take (5.15) v(x) = Cx 1 for some positive constant C.
On the other hand, (5.5) implies that the following inequality holds true for the function v j : By the uniform convergence in (5.14), we have that for any ε > 0 there is j 0 such that |Cx 1 − v j (x)| < ε whenever j > j 0 . Since ∂{v j > 0} is h 0 /2 thick in B 1 it follows that there is y j ∈ ∂{v j > 0} ∩ B 1 such that y j = e 1 h 0 /4 + t j e ′ , for some t j ∈ R, where e 1 is the unit direction of x 1 axis and e ′ ⊥ e 1 . Then we have that C h 0 4 − 0 = |v(y j ) − v j (y j )| < ε, which is a contradiction if ε is small. This finishes the proof of (5.4).
So we can extract a converging subsequence such that v j → v 0 uniformly in B 3 4 and ∇v j → ∇v 0 weakly in L p (B 3 4 ). As in the proof of Proposition 5.1 we get that ∆ p v 0 ∇v 0 = 0, and consequently v 0 is a p-harmonic function in R n .

Viscosity solutions
If Θ(u, x 0 , r) is either small or ∂{u > 0} is non-flat then u has linear growth near x 0 ∈ ∂{u > 0}. Thus the remaining case to be analyzed is the following : Θ(u, x 0 , r) is large and ∂{u > 0} is flat near x 0 .
To tackle this remaining case we want to use the stratification argument from [DK18] for the viscosity solutions in order to obtain the Lipschitz continuity of u. This will be done by combining the above results. To define the notion of viscosity solution we let Ω + (u) = {u > 0} and Ω − (u) = {u < 0}. If the free boundary is C 1 smooth then To justify the form of the free boundary condition we first show that for smooth free boundaries (7.1) is true.

Definition 7.2. Let Ω be a bounded domain of R N and let u be a continuous function in Ω. We say that u is a viscosity solution in Ω if
and Ω − (u), ii) along the free boundary Γ, u satisfies the free boundary condition, in the sense that: a) if at x 0 ∈ Γ there exists a ball B ⊂ Ω + (u) such that x 0 ∈ ∂B and for some α > 0 and β ≥ 0, with equality along every non-tangential domain, then the free boundary condition is satisfied for some α ≥ 0 and β > 0, with equality along every non-tangential domain, then The main result of this section is the following: Theorem 7.3. Let u ε j be a family of solutions to (P ε ), such that u ε j → u locally uniformly in B 1 . Then u is a viscosity solution in Ω in the sense of Definition 7.2.
The proof of Theorem 7.3, will follow from Lemmas 7.4 and 7.7 below. For the proof of Lemma 7.4 see Appendix [DK18].
Lemma 7.4. Let 0 ≤ u ∈ W 1,p (Ω) be a solution of ∆ p u = 0 in Ω and x 0 ∈ ∂Ω. Suppose that u continuously vanishes on ∂Ω ∩ B 1 (x 0 ). Then a) if there exists a ball B ⊂ Ω touching ∂Ω at x 0 , then either u grows faster than any linear function at x 0 , or there exists a constant α > 0 such that where ν is the unit normal to ∂B at x 0 , inward to Ω. Moreover, equality holds in (7.4) in any non-tangential domain. b) if there exists a ball B ⊂ Ω c touching ∂Ω at x 0 , then there exists a constant β ≥ 0 such that with equality in any non-tangential domain.
With this, we are able to prove Theorem 7.3 by utilizing the following anisotropic scaling argument.
Remark 7.6. From Theorem 7.5 it follows that the limit u is a viscosity solution in the sense of Definition 7.2.
We recapitulate the statement of Theorem 7.5 and amplify it by proving a more quantitative result. It can be proven in much the same way as Lemma 6.1. We give only the main ideas of the proof. Lemma 7.7. Let u ε j be a family of solutions to (P ε ), such that u ε j → u locally uniformly in B 1 . Let x 0 ∈ ∂{u > 0} and r > 0 small such that B r (x 0 ) ⊂ Ω. Assume that sup B r (x 0 ) u − ≤ C 0 r (resp. sup B r (x 0 ) u + ≤ C 0 r) ∀r ∈ (0, r 0 ), for some constant C 0 depending on x 0 and r 0 small.
Then there exists a constant σ > 0 such that sup B r (x 0 ) u + ≤ (1 + σC 0 )r (resp. sup B r (x 0 ) u − ≤ (1 + σC 0 )r). Proof. We will show only one of the claims, the other can be proved analogously. Suppose that and we claim that where S(k) := sup B 2 −k (x 0 ) |u|, for any k ∈ N. To prove this, we argue by contradiction and we suppose that (7.7) fails. Then there is a sequence of integers k j , with j = 1, 2, . . ., such that From the bound u ∞ ≤ 1 and (7.8) it follows that k j → ∞ as j → +∞. Also, notice that (7.8) implies that (7.9) σ j := 2 −k j S(k j + 1) ≤ 2 j → 0 as j → +∞.

Now, we introduce the scaled functions
S(k j +1) , for x ∈ B 1 . Then, from (7.6) and (7.9), it follows that Furthermore, it is not difficult to see that (7.8) implies that (7.11) sup B 1 |v j | ≤ 2, and sup The same compactness argument as in the proof of Lemma 6.1 gives that v j W 1,p (B 3 4 ) are uniformly bounded. Also, it implies (with the help of Proposition 2.6) that we can extract a converging subsequence such that v j → v 0 uniformly in B 3 4 and ∇v j → ∇v 0 strongly in L p (B 3   4 ). Moreover, (7.10), (7.11) and Theorem 4.1 give that , v 0 (0) = 0, and sup which is in contradiction with the strong minimum principle. This shows (7.7) and finishes the proof.
8. Lipschitz continuity of u: Proof of Theorem 1.1 Proposition 5.1 and Lemma 6.1 can be summarized by saying that if at x 0 ∈ ∂{u > 0} the free boundary is neither flat nor the set is {u < 0} is thick then we have uniform linear growth at x 0 . Thus we only have to look at those free boundary points where u < 0 is nontrivial, since in its complement we know that u is Lipschitz.
We begin by introducing another notion of flatness, suitable for the viscosity solutions, in terms of the ε−monotonicity of u. More precisely, we give the following definitions: Definition 8.1. We say that u ∈ C(B 1 ) is ε−monotone in B 1−ε if there are a unit vector e and an angle θ 0 with θ 0 > π 4 (say) and ε > 0 (small) such that, for every ε ′ ≥ ε, We denote by Γ(θ 0 , e) the cone with axis e and opening θ 0 .
Definition 8.2. Let u be a viscosity solution in B 1 (x), with x ∈ ∂{u > 0}. We say that u is ε−monotone in the cone Γ(θ 0 , e) if it is ε−monotone in any direction τ ∈ Γ(θ 0 , e). Furthermore, we say that u is ε-monotone in the cone Γ(θ 0 , e) in B r (x) if the function U(y) = u(x+yr) r , with y ∈ B 1 , is so in the cylinder B ′ where B ′ r denotes the ball with radius r of codimension 1.
One can interpret the ε−monotonicity of u as closeness of the free boundary to a Lipschitz graph with Lipschitz constant sufficiently close to 1 if we leave the free boundary in directions e at distance ε and larger. The exact value of the Lipschitz constant is given by tan θ 0 2 −1 . Then for suitable ε and θ 0 , which we call critical flatness constants, the ellipticity propagates to the free boundary via Harnack's inequality giving that Γ is Lipschitz. Furthermore, Lipschitz free boundaries are, in fact, C 1,α regular. Therefore we have Theorem 8.3. Let x 0 ∈ ∂{u > 0} such that h(x 0 , r) < rh 0 and Θ(u, x 0 , r) ≥ δ with δ > 0 as in Lemma 6.1. Then there is a constant C = C(n, M, δ, h 0 ) such that The proof is a slight modification of Theorem A, since the condition Θ(u, x 0 , r) ≥ δ implies that there is a negative phase and u is a viscosity solution.

Behaviour near free boundary
With Lipschitz continuity we can show that the results in [CLW97] hold for the nonlinear problem (P ε ). With minor modifications the following theorem follows from the results of [CLW97].
Theorem 9.1. Let u ε j be solutions to (P ε ) in a domain D ⊂ R n . Let x 0 ∈ D and suppose u ε j converge to u 0 uniformly on compact subsets of D as ε j → 0. Then the following holds The last two theorems exhibit the behavior of u near the free boundary Theorem 9.2. Let u ε j be solutions to (P ε ) in a domain D ⊂ R n such that u ε j → u uniformly on compact subsets of D and ε j → 0. Let x 0 ∈ D ∩ ∂{u > 0} and let γ ≥ 0 be such that Then, Proof. We have divided the proof into six steps: Step 1) Let α := lim sup |∇u(x)| By Theorem 8.3 u is Lipschitz continuous, therefore α is finite. If α = 0 then we are done. Thus let us assume that α > 0. There is a sequence x k ∈ {u > 0} such that x k → x 0 and lim k→∞ |∇u(x k )| = α. Denote d k = dist(x k , ∂{u > 0}), then we know that there is z k ∈ ∂{u > 0} such that d k = |x k − z k |.
Step 2) Let We have that |∇u d k (x)| = |∇u(z k + d k x)| ≤ C because u ∈ C 0,1 loc (D) by Theorem 8.3. Consequently, u d k (x) are uniformly bounded on compact sets of R n since u d k (0) = 0. Therefore there is a subsequence (still labelled u d k (x)) such that u d k → u 0 uniformly on the compact subsets of R n and the limit u 0 is Lipschitz continuous on the compact subsets of R n . Step Note that x k ∈ ∂B 1 and B 1 (x k ) ⊂ {u d k > 0}. Se can exract a subsequence, still labelledx k , such thatx k →x such that u 0 (x) ≥ 0 in B 1 (x) and ∆ p u 0 = 0 in B 1 (x).
We can also extract a converging subsequence from the sequence of unit vectors Note that ∇u(x k ) = ∇u d k (x k ). Hence it is enough to show check that To see this we first note that ψ(u d k − u d m ) ∈ W 1,p 0 (B 1 (x)) for given 0 ≤ ψ ∈ C ∞ 0 (B 1 (x)) for sufficiently large k, m. Therefore where the last inequality follows from a well know estimate (2.1) with γ depending only on n, p. Thus for an appropriate choice of ψ ≥ 0 we get from (9.4) that for every ball B satisfying 2B ⋐ B 1 (x) for sufficiently large k, m. Here we assume that 2B is the ball with the same center as B and of radius equal to the diameter of B. On the other where the last line follows from the β-Hölder estimate for gradient (see [DM93]) and (9.5). Since r is arbitrary and u d k − u d m L ∞ (B 2r (x)) → 0, if k, m are sufficiently large, it follows that that ∇u d k → ∇u 0 uniformly in some uniform neighborhood ofx. As result we get that and (9.2) follows.
Step 4) We claim that |∇u Since δ > 0 is arbitrary the claim follows. By a similar argument we can prove that |∇u − | ≤ γ.
Step 5) Let v = ∂u 0 ∂ν . Then differentiating ∆ p u 0 = 0 in ν direction we get that div(a(∇u 0 )∇v) = 0 in B 1 (x), where a(∇u 0 ) is a matrix with p-laplacian type growth. Since by (9.6) ∇u 0 0 nearx, it follows that v solves a uniformly elliptic equation in B R (x) for some R > 0 small. Since v attains local maximum atx then it follows that v = α in B R (x). For the sake of simplicity we assume that ν = e 1 , thus u = αx 1 + g(x ′ ), x ′ = (0, x 2 , . . . , x n ) for some function g. Form |∇u 0 | ≤ α it follows that g must be constant. From the unique continuation theorem [?] it readily follows that there is a pointx such that On the other hand from the asymptotic expansion [?] we have that there areᾱ,γ such that Step 6) To finish the proof we blow-up u 0 one more time. Let u 0λ (x) = 1 λ u 0 (x + λx). From Step 5 we conclude that for a subsequence these functions converge to u 00 = αx 1 + µx − 1 . From Proposition 2.9 it follows that there is a sequence ε 00 j such that u ε 00 j are solutions to (P ε ) and u ε 00 j → u 00 = αx 1 + µx − 1 . If µ = 0 then Theorem 8 (ii) gives (9.1). If µ > 0 then from Theorem 8 (iii). If µ < 0 then since ∇u 0λ k → ∇u 00 * -weakly in L ∞ loc and |∇u − | ≤ γ it follows that |µ| ≤ γ and we can apply Theorem 8 (i).
Theorem 9.3. Let u ε j be a solution to Pε j in a domain D j ⊂ R n such that D j ⊂ D j+1 and ∪ j D j = R n . Let us assume that u ε j converges to a function U uniformly on compact sets of R n and ε j → 0. Assume, in addition,that U ∈ Lip(1, 1) in R n and ∂{U > 0} = ∅. If γ ≥ 0 is such that |∇U − | ≤ γ in R n then, Proof follows from minor modifications of the previous one.

Application: Weak solutions
In this section we study the set of singular points of weak solutions, a subclass of variational solutions. We begin with the following [AC81], [Wei98] we give the definition of the weak solutions.
Proof. For a given test function φ ∈ C 0,1 (Ω; R n ) the set suppφ ∩ (∂{u > 0} \ ∂ red {u > 0}) is compact. Consider a covering of this set by B r i (x i ) such that and where to get the last line we used ∆ p u = 0 in E + j . Note that the integrals in above computation involving the second order derivatives of u are well defined thanks to the weighted local W 2,2 estimates for p−harmonic functions. Thus Since ∂{u > 0} \ F ⊂ ∂ red {u > 0}, on ∂ red {u > 0} we have u m = −|∇u|ν m and the free boundary condition (p − 1)(|∇u + | p − |∇u − | p ) = λ p is satisfied, we conclude Thus the integral over suppφ \ F is 0. The remaining integral F |∇u| p + λ p χ {u>0} divφ − p|∇u| p−2 ∇uDφ∇u ≤ (p + 1) Dϕ ∞ F |∇u| p tends to zero as δ → 0 since N(δ) i=1 r n i < δ 2 and we can utilize the absolute continuity of the integral. Sending δ to zero the result follows.

A BMO estimate
In [DK18] we have proven that the gradient of a minimizer is locally BMO provided that p > 2. A weaker estimate holds for the solutions of (P ε ). More precisely, in this section we show that the tensor T lm (∇u ε ) = p|∇u ε | p−2 u ε l u ε m − |∇u ε | p δ lm can be decomposed to a sum of divergence free and BMO tensors.
Note that in Bogovski's formula v has the form v(x) = Ω k(x, y) f (y) with a singular kernel k(x, y), and the derivatives of k behave like Calderon-Zygmund kernels [Gal11] III 3.15-3.17.
Let η be a cut off function of some ball B ⋐ B 1 . Localizing T ij we have that ∂ j (ηT ij ) = f i where f i = η∂ i B * + η i T ij , with f j = 0. Hence from Bogovski's formula and the estimates for the Calderon-Zygmund operators, we get that there is T ij ∈ BMO(B) such that where ∂ j (T 0 ij ) = 0.