The two membranes problem for fully nonlinear operators

We study the two membranes problem for two different fully nonlinear operators. We give a viscosity formulation for the problem and prove existence of solutions. Then we prove a general regularity result and the optimal $C^{1,1}$ regularity when the operators are the Pucci extremal operators. We also give an example that shows that no regularity for the free boundary is to be expected to hold in general.


Introduction
The two membranes problem was first studied by Vergara-Caffarelli [VC71] in the context of variational inequalities to describe the equilibrium position of two elastic membranes in contact with each other that are not allowed to cross. He considered the linear elliptic case, in which the problem can be reduced to the classical obstacle problem by looking at the difference between the two functions representing the position of each membrane.
Nearly 35 years later, Silvestre [Sil05] studied the problem for a nonlinear operator in divergence form. He obtained the optimal C 1,1 regularity of solutions together with a characterization of the regularity of the free boundary, that is the boundary of the set where the two functions coincide. The strategy in his proof was to show that the difference of the two functions satisfies an obstacle problem for the linearized operator, for which the regularity theory of the solutions and the free boundary are well known. An important remark is that in both of these cases the operator governing the behavior of each function is the same.
In a recent paper, Caffarelli, De Silva and Savin [CDS16] considered the two membranes problem for (possibly nonlocal) different operators, i.e. they consider the case in which one of the membranes (say the lower one) satisfies an equation that has higher order with respect to the other one. Here, heuristically, the lower order operator can be treated as a perturbation and some regularity for the lower membrane is obtained. Regularity from the upper membrane can then be deduced by solving an obstacle problem (with the lower membrane as obstacle) and obtaining estimates for solutions of nonlocal obstacle type problems in which the obstacle is not smooth. Repeating these arguments, the optimal regularity is achieved.
We also point out that the problem has been studied by several authors in the general case of N membranes, see [CV85], [CCVC05], [ARS05].
Here, motivated by a model from mathematical finance, we consider a version of the two membrane problem for two different fully nonlinear operators. It is worth pointing out that, for the case of two different operators of the same order, the only result available (to the best of the authors' knowledge) is the Hölder regularity obtained in [CDS16] (see the Introduction there for a discussion of the difficulties of this problem). In this paper we prove C 1,α regularity of the solution pair for (concave or convex) operators satisfying a sort of compatibility condition (see (1.4) below) and C 1,1 regularity for the case of the Pucci extremal operators, which is optimal. Moreover, we give an explicit example that shows that no regularity can be expected to hold for the free boundary in general.
1.1. Notation and preliminaries. Throughout this paper the ellipticity constants λ, Λ ∈ R will be fixed and will satisfy 0 < λ < Λ. Given these, we denote by M + and M − the Pucci extremal operators with respect to the class of symmetric matrices whose eigenvalues lie between λ and Λ, that is for any symmetric matrix X and where L λ,Λ = {A ∈ R n×n : A is symmetric and λId ≤ A ≤ ΛId} X ≥ Y meaning as usual that X − Y is a positive semidefinite matrix.
Recall that an operator F : R n×n → R is said to be uniformly elliptic with repect to the class L λ,Λ if it satisfies for any pair of symmetric matrices X and Y . We will assume without loss of generality that F (0) = 0. A useful remark that follows from (1.1) is that if u is a function satisfying F (D 2 u) = f then In particular, u is a subsolution and a supersolution of two (possibly different) elliptic equations with bounded measurable coefficients.
1.2. The two membranes problem for fully nonlinear operators. The problem we will consider is the following: given two functions u 0 , v 0 ∈ C γ (∂B 1 ) and f, g ∈ C γ (B 1 ) for some γ ∈ (0, 1), we want to study the solutions u and v of where Note that G thus defined will be concave and that particular examples are with Σ some set of indexes and A α ∈ L λ,Λ for every α. If A α can be any matrix in L λ,Λ then F = M + and G = M − . It is in this latter case that we prove the optimal regularity. Note that the strict inequality assumed for the ellipticity constants avoids these operators to become just a multiple of the Laplacian. Equation (1.3) is to be understood in the viscosity sense. More precisely: if ϕ is a C 2 function in B 1 satisfying for some (i.e. ϕ touches u by below at x 0 ) then Similarly if ϕ touches v by above and of course the opposite inequalities (last two equations in (1.3)) hold if ϕ touches u by above (or v by below) in Ω. A simple remark that will be useful is that it is equivalent to use paraboloids instead of general C 2 functions.
Note that the convexity of F as well as the Hölder regularity for f and g are natural assumptions if we want to get optimal regularity. In fact, one expect solutions to this problem to be C 1,1 as long as the equation they solve on the noncontact set is "good enough", meaning that we have at least C 1,1 regularity for it. This, in principle, is not true in general if F is not convex or f and g are merely bounded. Also, for the problem to make sense we will assume throughout the paper that u 0 > v 0 on ∂B 1 . Moreover, we will assume that f − g ≥ 0. Notice that if this was not the case the problem could loose interest and degenerate into just two independent fully nonlinear equations. Indeed, if f − g < 0, we would have (see (1.2)) and due to the maximum principle u > v in B 1 . Then there is no contact set and we just have the respective equations for u and v. Equation (1.3) models a so called "bid and ask" situation in which we have an asset, a seller (represented by u) and a buyer (represented by v). The price of the asset is random and the transaction will only take place when u and v "agree on a price", i.e. when u = v. Moreover, we want to model the expected earnings of u and v, assuming that their strategy is optimal.
One can think of this problem as having two different (although related) features: on one hand, we have an "obstacle type" situation, in which u tries to maximize gain with v being an obstacle and vice versa (v minimizing cost and u being an obstacle), hence the constraint u ≥ v. But perhaps more interesting is the special relation between u and v. Because of the "bid and ask" nature of the model, the Bellman type equations that govern the behavior of our solutions are closely related (recall F (X) = −G(−X)) and it is precisely this feature which opens a way to get regularity even though the operators are different.
The main result of this paper is the following:

Existence
In this section we prove existence of solutions for our problem. We use the method of penalization, i.e. we are going to consider a family of unconstrained "penalized equations" whose solutions are uniformly bounded in some Hölder space and hence convegent up to a subsequence. Then we are going to show that the limit of that subsequence is actually a solution to (1.3) (see [KS00]).
The penalized problem we are going to consider is the following: where for each ǫ > 0 we define and To get the existence and a priori bounds for solutions of (2.1) we will use a fixed point argument. Hence, we will need global regularity results for equations of the form where H is a uniformly elliptic operator. Here we mostly follow Chapter 4 of [CC95] but since the proofs need to be modified slightly we sketch them below for completeness: Proposition 2.1. Let u be a viscosity solution of (2.5) with h ∈ L ∞ (B 1 ) and u 0 ∈ C γ (∂B 1 ). Then for any x 0 ∈ ∂B 1 we have where C is a constant depending only on n, λ, Λ, γ, u 0 C γ (∂B 1 ) and h L ∞ (B 1 ) .
Proof. We translate, rotate and add a constant to u so that it is defined on For this let us define the barrier ψ(x) = Cx on ∂B and hence there. On the other hand, ψ satisfies in B in the viscosity sense if we take C large enough. From the maximum principle it follows that u ≤ ψ in B. Symmetrically, we see that u ≥ −ψ on ∂B and H(D 2 (−ψ)) ≥ h L ∞ (B 1 ) in B, so using the maximum principle again we get u ≥ −ψ and hence (2.7). Now we prove the global Hölder estimates.

Proof.
We start by recalling that by interior estimates (Proposition 4.10 in [CC95]) solutions of (2.5) are in C α loc (B 1 ) for some α > 0. Let η = min{γ/2, α}, x 1 , x 2 ∈ B 1 r = |x 1 − x 2 |, and take x ′ 1 , x ′ 2 ∈ ∂B 1 such that We assume without loss of generality that d 2 ≤ d 1 and we want to show that (2.9) We split the proof in two cases: Case 1: When r ≤ d 1 2 , from the rescaled version of the interior estimates applied to u − u(x ′ 1 ) on B d 1 (x 1 ) and our estimates at the boundary (2.6) we have as desired in this case. Case 2: When r > d 1 2 , again by the boundary estimates (2.6) and the triangle inequality In our next proof we are also going to use global Hölder estimates for "equations with bounded measurable coefficients": for α a positive constant and u 0 ∈ C γ (∂B 1 ). Then where C is a constant depending only on n, λ, Λ, u 0 C γ (∂B 1 ) and α and η ≤ γ/2.
Proof. The proof follows exactly as that of Proposition 2.2. We just point out that in order to prove the boundary estimates (2.6) a barrier argument for the Pucci extremal operators is used that is trivially adapted to a situation like (2.10). This is then combined with interior estimates that also hold for (2.10) (see [CC95]) to give (2.11) Finally, the following observation is going to be useful: it follows from the proof of Proposition 2.1 that the dependence of the constant on h L ∞ (B 1 ) is continuous. The same is true for the interior estimates (again, see [CC95]) and hence for the constant in (2.11).
We can now prove existence of the penalized problem: in the viscosity sense.
Proof. We will use Schauder's Fixed Point Theorem (see [GT98]). Letα = η/2 with η as in Proposition 2.2 and consider the map (2.13) Such u and v exist by Perron's method. Also, let with C to be determined later.
If we can show that X is a compact convex set in C η (B 1 ) × C η (B 1 ), that T is continuous in X and T (X) ⊂ X then by Schauder's Fixed Point Theorem there exists a solution to (2.12). We divide the proof in several steps: Step 1: convexity and compactness of X. The convexity is trivial. As for the compactness, it is a straight forward consequence of the Arsela-Ascoli theorem sinceα < η.
Step 2: T (X) ⊂ X. Notice that ifū,v ∈ Cα(B 1 ) and we let we have, as f, g, β are bounded, that h, k ∈ L ∞ (B 1 ). Hence, from Proposition 2.2 we know that for some constant C > 0 that depends only on n, λ, Λ, f L ∞ (B 1 ) , g L ∞ (B 1 ) , u 0 C γ (B 1 ) and v 0 C γ (B 1 ) (this is the constant C used to define X). In particular, this implies that (u, v) ∈ X.
Step 3: T is continuous.
We now show the main result of this section, which is the existence of solutions of the two membranes problem (1.3).
Theorem 2.5. There exist u, v ∈ C(B 1 ) that solve (1.3) in the viscosity sense.
We first want to see that u ≥ v. Assume not, i.e. suppose the exists x ∈ B 1 such that In particular, u ǫ − v ǫ has to have a negative minimum at some point y ∈ B 1 (recall that on ∂B 1 we have u 0 ≥ v 0 ). Moreover, u ǫ − v ǫ satisfies, by the convexity of F and the fact that F (X) = −G(−X), in the viscosity sense. Let P be a plane touching uǫ−vǫ 2 by below at y. Then F (D 2 P ) ≡ 0 but since in particular P is a C 2 function we must have F (D 2 P ) ≤ 1 2 (f (y) − g(y) + 2β ǫ (P (y))) < 0 a contradiction. Now we want to show that u and v satisfy the corresponding equations. Let us start by showing that F (D 2 u) ≤ f in B 1 .
Let ϕ be a paraboloid touching u by below at x 0 ∈ B 1 . Given ξ > 0 there exists Also, for any η > 0 we can choose ǫ small enough so that a translation of ϕ(x) − η 2 |x − x 0 | 2 (which we callφ) touches u ǫ by below at Since ξ was arbitrary we get and letting η → 0 we get as desired (recall that F is continuous in the space of matrices).
Using again the uniform convergence, the definition of viscosity solutions and considering a test function ϕ that touches u by above we can similarly show that F (D 2 u) ≥ f in B 1 ∩ Ω and conclude that F (D 2 u) = f in B 1 ∩ Ω. The proofs of G(D 2 v) ≥ g in B 1 and G(D 2 v) = g in B 1 ∩ Ω are analogous to the previous reasoning. It is immediate from uniform convergence that u = u 0 and v = v 0 on ∂B 1 .
Remark 2.6. It is noting that the proof would still hold if we slightly relax the assumptions on the operators by asking just F (X) ≤ −G(−X).
Remark 2.7. We point out that there is no uniqueness in this problem. This comes from the fact that "there is no equation" on the contact set. In fact, let (for n = 1) u and v are C 1,1 functions and they are strong solutions (and hence viscosity solutions) of (1.3) with F = M + , G = M − , f = Λ and g = −Λ. However, we can make a perturbation ψ ∈ C ∞ c ((−1, 0)) such that −1 ≤ ψ ′′ ≤ 1 in (−1, 0) and get another solution.
Of course this example can be easily generalized to n ≥ 2 choosing and modifying the right hand sides accordingly.
However, uniqueness does hold in the "nonexcercise region" Ω. In fact, if two pairs of solutions (u, v) and (u ′ , v ′ ) satisfy by the maximum principle for fully nonlinear elliptic equations (notice that in B 1 ∩Ω we have F (D 2 u) = f and G(D 2 v) = g).

Regularity for the solution pair
In this section we prove our main regularity result, Theorem 1.1. The fact that this is the optimal regularity can be easily seen by considering u(x) = To prove Theorem 1.1 we show that solutions to (1.3) satisfy the hypothesis of the following Theorem (see Theorem 2.1 in [IM16]): for some open set Ω ⊂ B 1 and some elliptic operator F that is either concave or convex. Then, there exists a constant C depending only on f C γ (B 1 ) , u W 2,n (B 1 ) , the dimension and the ellipticity constants such that The first step is to show the following Calderón-Zygmund type estimate: Proposition 3.2. Let u and v solve (1.3). Then u and v belong to W 2,p (B 1/2 ) for any 1 < p < ∞ and for some constant C depending only on n, λ, Λ, Proof. We prove the result for u, the proof for v is analogous. We will show that |F (D 2 u)| ≤ C in the viscosity sense for some universal constant C. The result will then follow from Theorem 7.1 in in [CC95] (recall that F (·) is a convex operator). Let ϕ be a C 2 function touching u by below at x 0 ∈ B 1 . Recall that u is a viscosity supersolution across the whole ball (disregarding if x 0 is in the contact set or not), so we have If instead ϕ touches u by above, we separate two cases: Case 1: if x 0 ∈ Ω then u is also a subsolution and we get ∈ Ω, notice that ϕ also touches v by above, and v is a subsolution for G across the whole ball. Then But for any symmetric matrix X we have G(X) ≤ F (X). Thus and we are done.
Remark 3.3. Notice that the proof is still valid if we just require F (X) ≥ G(X). Now we show that when problem is given by the Pucci extremal operators solutions are C 1,1 on the contact set (i.e. they have bounded second derivatives). More precisely: Proof. By Proposition 3.2 u and v are W 2,p functions in, say, B 3/4 so we only need to show an almost everywhere bound on D 2 u and D 2 v in Ω c . Also, since the notions of viscosity solution and strong solution coincide for W 2,p with p ≥ n (see [CCKS96]) we have that (1.3) is satisfied a.e. If x is a point in the interior of Ω c for which (1.3) is satisfied, u and v coincide in a neighborhood of x and hence, letting e u and e v denote the eigenvalues of D 2 u and D 2 v respectively, we find If x ∈ ∂Ω c (again, a point at which (1.3) holds), u − v has a minimum at x and hence D 2 (u − v)(x) is nonnegative definite, which in particular implies ∂ ee u(x) ≥ ∂ ee v(x) for any direction e ∈ S n−1 . Let us now pick a system of coordinates, say {e 1 , . . . , e n }, in which D 2 v(x) is diagonal. Moreover let us assume without loss of generality that the first m eigenvalues of D 2 v are nonpositive and the remainig n − m positive. Let then A be a diagonal matrix with λ in the first m positions of its diagonal and Λ otherwise. Since A is a competitor in the sup and inf that define F and G respectively we have, using the equation, |e v | so we get the bound for D 2 v(x). The proof of the bounds for D 2 u(x) is completely analogous.
Finally, we can give the Proof of Theorem 1.1. Again, we prove the result for u. Notice that, due to Proposition 3.2, u is W 2,n (B 3/4 ). Moreover, by Proposition 3.4 the Hessian of u is bounded a.e. inside the contact set in B 1/2 , and hence we have F (D 2 u) = f (x) in B 1/2 ∩ Ω |D 2 u| ≤ C in B 1/2 ∩ Ω c and we can apply Theorem 3.1 to get that u ∈ C 1,1 (B 1/4 ) as desired.

Free boundary
The classic approach to study the regularity of the free boundary of the double membrane problem consists on substracting the two membranes (solutions), say w := u − v, and reduce the situation to an obstacle-type problem (note that w thus defined is nonnegative). One of the key steps of the analysis of the free boundary is to show that w satisfies a non-degeneracy property around free boundary points; that is, given x 0 ∈ ∂{w > 0} one wants to show that sup ∂Br(x 0 ) w ≥ Cr 2 for r > 0 (4.1) where C > 0 is a universal constant. In the case of (1.3) this property is not satisfied. Indeed, let C be any positive constant and consider u(x, y) := x 2 − y 2 + Cx 3 + and v(x, y) := x 2 − y 2 .
Here x + = max{x, 0}. Notice that In particular u, v solve (1.3), 0 ∈ ∂{w > 0} and sup ∂Br(0) w = C sup ∂Br(0) x 3 + = Cr 3 < Cr 2 (4.2) for any r < 1. In fact, by previous the following example we can see that no free boundary regularity can hold in general. If we make u(x, y) := x 2 − y 2 + ψ(x, y) and v(x, y) := x 2 − y 2 with ψ a nonnegative smooth function we can make the contact set arbitrarily bad and still get solutions of (1.3).