A MINIMIZATION PROBLEM WITH FREE BOUNDARY FOR p -LAPLACIAN WEAKLY COUPLED SYSTEM

A bstract . In this paper we consider a weakly coupled p -Laplacian system of a Bernoulli type free boundary problem, through minimization of a corresponding functional. We prove various properties of any local minimizer and the corresponding free boundary.

for some constants Q min and Q max .We are interested in regularity properties of minimizers u, as well as the free boundary . In fact, Δ u p i is a nonnegative Radon measure with support on free boundary, Γ.The problem is to find a reasonable representation of this measure and put it into some partial differential equations context for further analysis.
This problem is referred to as Bernoulli-type free boundary problem, and is well studied in the literature, for the scalar case and for = p 2, starting with seminal work of Alt and Caffarelli [2], and also for any < < ∞ p 1 in in the work of Danielli and Petrosyan [8].There are very few results for Bernoulli-type problems that involve systems [7,11,20].Caffarelli et al. [7] studied the minimum problem (1) for = p 2 and show the smooth- ness of the regular part of free boundary as well as some partial result for Hausdorff dimension of singular part.Indeed, they apply a reduction method to reduce the problem to its scalar counterpart and the same result for the scalar case can be extended to the vectorial problem.Also, a vectorial Bernoulli problem with no sign assumption on the components was studied by Mazzoleni et al. [20].In the study by De Silva and Tortone [11], the same result has been obtained by the viscosity approach and improvement of flatness.
In this article, we deal with a (weakly coupled) cooperative system for p-Laplacian version of Bernoulli- type problem, following similar procedure as that in the study by Caffarelli et al. [7].
Remark 1.1.It should be remarked that our approaches in this article, with some extra efforts, can be adapted to variable exponent case, as well as variable coefficient one.Similar types of results are then expected.

Notation
For clarity of exposition, we shall introduce some notations and definitions, which are used frequently in this text.
Throughout this article, n will be equipped with the Euclidean inner product ⋅ x y and the induced norm

Plan of the article
The article is organized as follows: In Section 2, we study the existence of minimizer (Theorem 2.1) and show that minimizers are p-subharmonic (Lemma 2.3).Section 3 is devoted to the regularity property of solutions, including the Hölder regularity (Lemma 3.2) and the Lipschitz regularity (Theorem 3.5).Section 4 consists of the proof of nondegeneracy property (Lemma 4.1).Also, in Theorem 4.2, an estimate for the density of the free boundary is obtained, which is enough to prove that the free boundary has zero Lebesgue measure.The vector-valued measure Δ u p (Theorem 5.1) and − n 1 ( )-Hausdorff dimension of free boundary (Theorem 5.2) are discussed in Section 5.The main result in Section 6 is the flatness of regular part of the free boundary (Theorem 6.5).We prove a partial result for the regularity of free boundary in Section 7 (Theorem 7.5), along with C α 1, -regularity of the free boundary when p is sufficiently close to 2 (Theorem 7.6).In Appendix A, we deal with nontangentially accessible (NTA) domain properties of the free boundary.Also, in Appendix B, we present an auxiliary lemma to study the asymptotic behavior of p-harmonic functions.
2 Existence of a minimizer Theorem 2.1.If < ∞ J g ( ) , then there exists an absolute minimizer of J over class .
Proof.Obviously, the functional is nonnegative, and hence it takes an infimum value.Let u k be a minimizing sequence ), and up to a subsequence, we can assume that a.e. in Ω, for some ∈ u .The latter convergence implies and the weakly lower semicontinuity of the norm implies that Although all results in this article are proved for local minimizers, for the sake of convenience, we argue with absolute minimizers.
and consequently (by the maximum principle), Then, ∈ v , and we can choose v as a competitor, so The second statement in the lemma relies on the strong minimum principle for p-harmonic functions.Let for a component and some interior point of ∈ x Ω 1 , we obtain by virtue of the strong maximum principle that ≡ u 0 i in Ω 1 .□ 3 Regularity of local minimizers Lemma 3.1.Let u be a (local) minimizer of J, and v i be the harmonic replacement (majorant) 1 for u i in ⊂ B Ω (for B a small ball).Then, there is a universal constant for all ≠ j i and extend v i by u i in ⧹B Ω .If B is small enough (when u is absolute minimizer, we do not need this assumption), then we have ( ), and consequently, where for the third equality, we have used which shows the desired estimate for ≥ p 2.
In case < ≤ p 1 2, we have On the other hand, using the Hölder inequality, we have We conclude the proof by applying ∫ ∫ .
On the other hand, if v i is the p-harmonic replacement of u i inside B r , we have the gradient estimate (see [17]) Now, let us take some < ∕ ρ r 2, which will be specified below, and apply Lemma 3.1 in B y r ( ), ) sufficiently small, we obtain By virtue of Morrey's theorem [19] we conclude the proof of the lemma.□ The next lemma is essential to prove the Lipschitz regularity of the minimizers.
We need to remark that the constant C is independent of the boundary values of u on ∂Ω.In other words, when going away from a free boundary, but staying uniformly inside the domain Ω, the minimizer cannot grow too large, regardless of the boundary values.In other words, for large enough boundary values, the origin cannot be a free boundary point.
Proof.For the sake of convenience, consider = i 1. Towards a contradiction, assume that there is a sequence of bounded solutions and define . We have also , and therefore, m k is attained at some . By the Harnack inequality for p-harmonic functions, there is a constant = c c n p , ( ) such that In particular, We define the sequence Moreover, w k is a minimizer of . Now consider v k 1 to be p-harmonic replacement of u k 1 in ∕ B 3 4 and apply Lemma 3.1, when ≤ p 2 .Similar statement holds for < ≤ p 1 2, and we note that ∇w k L p-subsolution and uniformly bounded in B 1 ).Furthermore, w k 1 and v k 1 are uniformly C α in ∕ B 5 8 , and we can extract a subsequence (still denoted by w k . Hence, w 0 is also p-harmonic, and by the strong maximum principle, . On the other hand, equation ( 3) necessitates which is a contradiction.□ A direct consequence of the above lemma is the following estimate.
where C is the constant defined in Lemma 3.3.
With the above two results, we will obtain uniform Lipschitz regularity for minimizers.
Theorem 3.5.Let u be a (local) minimizer in Ω, then u is Lipschitz.Moreover, for every Once again, we remark that the constant C does not depend on the boundary values of the minimizer, as long as we stay uniformly inside the domain.

Proof. Step 1:
We show that u i is bounded in K with a universal constant C depending on the following

(
), and for any arbitrary point ∈ x K, there is a A minimization problem with free boundary for p-Laplacian weakly coupled system  7 sequence of points = ∈ x x x K ,…, k 0 with (we can assume K is connected, otherwise replace it with a bigger one which is connected) 1, by virtue of Harnack's inequality, there is a constant c such that Thus, Step 2: Here, we find a control on ∇u i at points close to Let us define . By p-Laplacian estimate for gradient, we obtain Step 3: , by the result of Step 2, we have already ∥ ∥ is universally bounded by the result of Step 1.Thus, ∇u x i | ( )| will be universally bounded.□ A straightforward corollary to this theorem, that can be useful later, is the following: Corollary 3.6.Let u be a (local) minimizer for our functional.For every Proof.Without loss of generality, we may assume = r 1.
On the other hand, Comparing with equation ( 5), we arrive at Therefore, if M is small enough, we obtain that = u 0 in B κ .□ An immediate consequence of the above lemma is the following.For any ⋐ K Ω, there are positive constants c 0 and C 0 such that if Theorem 4.2.For ⋐ K Ω, there exists a constant < = < c c n m p K 0 , , , ,Ω ) such that for any (local) minimizer u and for any (small) ball A minimization problem with free boundary for p-Laplacian weakly coupled system  9 Proof.By Lemma 4.  ( ).This gives the lower estimate in equation (7).To prove the estimate from above we assume, for simplicity, = r 1 and suppose (toward a contradiction) that there is a sequence of minimizers Since u k i and v k i are both uniformly Lipschitz in ∕ B 1 4 , we may assume that → u u and equation (8 , and from the strong minimum principle (since . On the other hand, from nondegeneracy property, Lemma 4.1, we know which implies a similar inequality for u 0 , and hence a contradiction.□ which, in virtue of Lemma 2.3, is a bounded nonnegative measure, i.e., a Radon measure.Obviously, λ i is the formal way of expressing Δ u p i in Ω.
Since each u i is p-subharmonic in Ω and ≥ u 0 i , we have that λ i is a positive Radon measure.Because u i is also p-harmonic in Theorem 5.1.For any ⋐ K Ω, there exist constants > c C , 0 such that for any (local) minimizer u, where, in the last inequality, we have used that u is Lipschitz.Letting ε tend to zero, we arrive at To prove the estimate from below, we argue indirectly.It also suffices to consider the case = r 1. Assume there is a sequence of minimizers u k in the unit ball B 0 1 ( ) Since the functions u k are uniformly Lipschitz continuous, we may assume that → u u , where u 0 is Lipschitz continuous as well.We may also extract a subsequence (still denote by Suppose this is true, then for every positive test function and u i 0 is p-harmonic for all = i m 1,…, (note that u i 0 is the limit of a sequence of p-subharmonic functions, and we already know that it is p-subharmonic).Since ≥ u 0 i 0 and = u 0 0 i 0 ( ) , by the minimum principle, we have ≡ u 0 i 0 in ∕ B 1 2 .On the other hand, by nondegeneracy property (Lemma 4.1) and that Therefore, a similar inequality holds for u 0 , and we arrive at a contradiction.
To close the argument, we need to prove equation (9).In fact, if } for sufficiently large k and u k i are p-harmonic in B ρ for all = i m 1,…, , (see Lemma 2.3).There- fore, one can extract a subsequence of u k locally converging to u 0 in C B α ρ for any ball ( ) and sufficiently large k.Passing to the limit, we obtain the same inequality for u 0 ,

∥ ∥ ( ( ))
A minimization problem with free boundary for p-Laplacian weakly coupled system  11 This along with the Lipschitz continuity of u 0 is enough to prove that

3). □
The next theorem follows easily from Theorem 5.1.The proof is the same as the proof of Theorem 4.5 in the study by Alt and Caffarelli [2].
Theorem 5.2.Let u be a (local) minimizer in Ω.Then, . (ii) There exist nonnegative Borel functions q i such that is a Borel measure and the total variation μ u | | is a Radon measure.We define the reduced boundary of A by where ν x u ( ) is the unique unit vector with if such a vector exists, and = ν x 0 u ( ) otherwise.For more details, see Chapter 4 in the study by Federer [14].

Local analysis
To proceed, we will need some properties of the so-called blow-up limits.
Lemma 6.1.Let u be a (local) minimizer in Ω, ⋐ K Ω, and , and . Consider the blow-up sequence For a subsequence, there is a limit u 0 such that For the proof, we refer to the works of Alt and Caffarelli [2] and Wilhelm Alt et al. [3].□ The following lemma shows that the blow-up limit is a minimizer in any ball.
where the positive part is taken separately for each component.We also have (owing to the Lipschitz continuity of u) and the convergences in equations ( 11) and ( 13), the limit of the left-hand side will be J u 0 0 ) and u is an absolute minimizer.Then, Proof.Let us define One can show that for sufficiently small t

( ) ( ( ))
A minimization problem with free boundary for p-Laplacian weakly coupled system  13 It follows that where By a change of variables, we have We also have Now differentiate J u t ( ) with respect to t and note that its minimum is attained at = t 0, then , we arrive at the desired claim, in the lemma.□ Definition 6.4.The upper where − ω n 1 denotes the volume of the unit sphere in − n 1 .We already know (see, e.g., Theorem 2.7 in [13]) that q q x o r as r d , 0 , for some vector = α α a , …, ) such that Proof.Without loss of generality, we assume that = ν x e n 0 ( ) .Let u k be a blow-up sequence with respect to balls B x r 0 k ( ), with blow-up limit u 0 .Since ν x 0 ( ) is the normal vector to , a s 0 .

B x
x x x ν x n u 0 : 0 This along with equation ( 13) implies } almost everywhere in n .By Lemma 6.2, we know that u 0 is an absolute minimizer of J 0 and so continuous.Then, where we have used assumption (14) and property (12).Therefore, for any test function

| | ( ) { }
A minimization problem with free boundary for p-Laplacian weakly coupled system  15 in the classical sense.We need to show that To see this, define w 0 by i n i n n It is obvious that w 0 is p-harmonic in whole n , as well as By Liouville's theorem, we conclude that w 0 is a linear function.The boundary value on = . This proves equation ( 16) and shows that We just have to prove equation (15).To do this, note that u 0 is an absolute minimizer of J 0 .By applying Lemma 6.3 for u 0 and some ).We say that the boundary condition ∇ Lemma 7.2.Let u be a (local) minimizer, then the boundary condition holds in the viscosity sense.
Proof.We show that the boundary condition holds on every point and up to a subsequence, we have If ∇ = ϕ x 0 0 ( ) , the viscosity condition holds trivially.Otherwise, the noncoincidence set On the other hand, u 0 is minimizer of J 0 (Lemma 6.2) and by Lemma 2.3, every nontrivial component of u 0 , say u i 0 , is positive in for some α i .Thus, any blowup of u 0 at = x 0 must be of the form ) .Again by applying Lemma 6.2 along with Lemma 6.3, we obtain Thus, equation ( 17) yields and so The same argument holds when  ( ), for some < ≤ r r 0 0 , is C α 1, for a universal exponent < < α 0 1.
Proof.We may assume that > u 0 Now returning to u, we obtain ≥ + ∕ u x α μ x 2 n ( ) ( ) in ∕ + B γ r k .This is a contradiction with the definition of k ℓ when k is sufficiently large.□ x | |, and B x r 0 ( ) will denote the open n-dimensional ball with center x 0 , radius r, and its boundary with ∂B x For the target space, m , we use several norms as follows: For convenience, we denote the Euclidean norm without the index, = u u 2 | | | | .We also use the Euclidean norm in the definition of ∞ L -norm, i.e.,

r 4 NondegeneracyLemma 4 . 1 . 1 ,
contains a free boundary point, then by Theorem 3.5, ≤ u Cr i on ∂B r .□ For any < < κ 0 there exists a constant = > c c κ n m p Q , , , , 0 min ( ) such that for every minimizer u and for any (small) ball ⊂

Remark 4 . 3 .
Theorem 4.2, along with the Lebesgue density theorem implies that the free boundary has zero Lebesgue measure be a test function and define the measure λ i by Also, we define the tangent cone of ∂ > u 0 {| | } at x 0 , denoted by ∂ > x u Tan 0 , 0 ( {| | } ), the set of all ∈ ν n such that for every > ε 0, there exist ∈ ∩∂ > If, in addition, x 0 is a Lebesgue point for Radon measure ⌊∂ ball with radius R (R is arbitrary and fixed).

1 .
Let ∈ C u Ω, m ( We denote the set of all regular points by u .Theorem 6. {| | }.First, we show that there is a Hölder function ∩ for some < ≤ r r 0 0 and < ≤ c 0 1 such that u 1 is a viscosity solution to the problem we know that this process cannot continue indefinitely without stepping out of B x r 0 ( ).So, we stop at the first k for which ⊄ for the universal constant γ.It necessitates that ⊂ w 0 is p-harmonic with boundary data = μ and γ.Thus, for ∈ + B γ , where1denotes length or the one-dimensional Hausdorff measure.