ON STRICT CONVEXITY AND CONTINUOUS DIFFERENTIABILITY OF POTENTIAL FUNCTIONS IN OPTIMAL TRANSPORTATION

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1 ON STRICT CONVEXITY AND CONTINUOUS DIFFERENTIABILITY OF POTENTIAL FUNCTIONS IN OPTIMAL TRANSPORTATION Neil Trudinger Xu-Jia Wang Centre for Mathematics and Its Applications The Australian National University Abstract. This note concerns the relationship between conditions on cost functions and domains, the convexity properties of potentials in optimal transportation and the continuity of the associated optimal mappings. In particular, we prove that if the cost function satisfies the condition (A3), introduced in our previous work with Xinan Ma, the densities and their reciprocals are bounded and the target domain is convex with respect to the cost function, then the potential is continuously differentiable and its dual potential strictly concave with respect to the cost function. Our results extend, by different and more direct proof, similar results of Loeper proved by approximation from our earlier work on regularity. 1. Introduction We continue our investigation on the regularity of potential functions in the optimal transportation problem [MTW, TW]. In this paper we prove strict convexity and C 1 regularity of potential functions for non-smooth densities. The strict convexity and C 1 regularity for solutions to the Monge-Ampere equation were established by Caffarelli [C1]. For the reflector design problem, which is a special optimal transportation problem [W2], analogous results were obtained in [CGH]. For more general optimal transportation problems, the C 1,α regularity has been obtained by Loeper [L]. In this paper we prove the strict convexity for potential functions, and obtain the C 1 regularity, through a different and more direct approach. We also use our preliminary analysis to plug a gap in [MTW] pertaining to the use of a comparison argument in the proof of interior regularity. Let Ω, Ω be two bounded domains in R n, and f, g be two nonnegative integrable functions on Ω, Ω satisfying the mass balance condition (1.1) Ω f = g. Ω Let (u, v) be potential functions to the optimal transportation problem, namely (u, v) is a maximizer of (1.2) sup{i(ϕ, ψ) : (ϕ, ψ) K}, This work was supported by the Australian Research Council 1

2 where (1.3) I(ϕ, ψ) = f(x)ϕ(x) + g(y)ψ(y), Ω Ω K = {(ϕ, ψ) C 0 (Ω) C 0 (Ω ) : ϕ(x) + ψ(y) c(x, y)}. We assume that the cost function is smooth, c C 4 (R n R n ), and satisfies conditions (A1)-(A3) below. It is known, (see [C4, GM, V] that there is a maximizer (u, v) to (1.2), which is also unique up to a constant if f, g are positive. The potentials (u, v) are semi-concave and satisfy (1.4) u(x) = inf y Ω {c(x, y) v(y)}, v(y) = inf {c(x, y) u(x)}. x Ω The optimal mapping T : x Ω y Ω can be determined, a.e. in Ω, by (1.5) Du(x) = D x c(x, y), where y also attains the infimum in (1.4). If u C 2 (Ω), it satisfies the Monge Ampère type equation (1.6,) det(dx 2 c f D2 u) = detc i,j g T in Ω, where c i = xi c, c i,j = xi yj c. The main result of this note is Theorem 1.1. Suppose that Ω is c-convex with respect to Ω, and the cost function c satisfies (A1)-(A3). Then the potentials u and v are fully c-concave. Moreover, if the densities f, g, together with their reciprocals are bounded, then v is strictly c-concave and u is C 1 smooth. We refer the reader to 2.1 for definitions of the relevant convexity and concavity notions relative to cost functions. By approximation and the uniqueness of potential functions (when f, g > 0), the boundedness conditions can be weakened to 0 f g T < C and f, g > 0. From the C 1 smoothness of u it follows that the optimal mapping T is continuous and is a homeomorphism if both Ω and Ω are c-convex. In addition, if f and g are C 1,1 then T is a diffeomorphism by [MTW]. The continuous differentiability of the potential u was proved originally by Loeper [L], using approximation by smooth solutions to the optimal transportation problem in [MTW]. Our correction to [MTW] also restores his result in its full generality. However our proof here is more direct, being independent of the interior regularity in [MTW]. We also provide extensions to more general domains of other results in [L], that were obtained by approximation from our global regularity results in [TW], namely the full c-concavity of both u and v. See also the remarks in 3. Our proof is also completely different from that in [C1], in which Caffarelli proved the strict convexity and C 1 regularity for 2

3 solutions to the Monge-Ampere equation with constant boundary condition. For higher regularity, the interior and global C 2,α estimates for solutions of (1.6), in the case of the quadratic cost function, were established in [C2, C3, U1], and earlier in [D] for n = 2. For cost functions satisfying (A1)-(A3) below, as already indicated, the interior and global regularity of potential functions was obtained in [MTW, TW]. The assumptions (A1)-(A3) are as follows. For some open set U, in R n R n, containing Ω Ω, we have (A1) For any (x, y) U, (p, q) D x c(u) D y c(u), there exists unique Y = Y (x, p), X = X(q, y), such that D x c(x, Y ) = p, D y c(x, y) = q. (A2) For any (x, y) U, det{c i,j (x, y)} 0. (A3) For any (x, y) U, and ξ, η R n with ξ η, (1.8) i,j,k,l,p,q,r,s (cp,q c ij,p c q,rs c ij,rs )c r,k c s,l ξ i ξ j η k η l c 0 ξ 2 η 2, where c 0 is a positive constant, and (c i,j ) is the inverse matrix of (c i,j ). Concerning the set U, we assume that the convex hulls of the sets D x c(x, Ω ), D y c(ω, y) lie in D x c(u), D y c(u) respectively for each x Ω, y Ω, which is automatically true, for example, if Ω, Ω are c-convex. We remark that condition (A3) is also symmetric in x and y and that inequality (1.8) is equivalent to (1.9) pk p l c ij (x, Y )ξ i ξ j η k η l c 0 ξ 2 η 2 for all (x, Y ) U and ξ η R n, where Y = Y (x, p) is given by (A1), which is smooth in x and z by (A2). Note also that for global regularity in [TW] and the subsequent application in [L], condition (A3) can be relaxed to its degenerate form, c 0 = 0, called A3w in [TW]. We divide the proof of Theorem 1.1 into several short sections. We first introduce in 2.1 various notions of convexities and concavities relative to the cost function c. We then indicate in 2.2 a geometric property of (A3); (see also [L]). In 2.3 we give an analytic formulation of the c-convexity of domains, corresponding to that in [TW]. A geometric characterization of c-convex domains (under condition A3) is given in 2.4. In 2.5 we prove that a local c-concave function is fully c-concave if the domain is c-convex and the cost function c satisfies (A3). We also prove the connectedness of the contact set under the same hypotheses, which is a key ingredient in the proof of Theorem 1.1. As remarked there, these arguments also extend to the weaker condition (A3w). The resultant properties are formulated as Theorem 2.1. We also indicate their application to smooth functions which are degenerate elliptic for equation (2.6),(Corollary 2.2) and to the minimum of two c-functions, (Corollary 2.3). In 2.6 we prove that the potential functions are locally c-concave under condition (A3) and conclude the full c-concavity of u and v under the hypothesis that either domain is c-convex. We also show that u is C 1 if and only if v is strictly c-concave. We then complete the proof of Theorem 1.1 in 2.7. Various remarks are given in 3. 3

4 We are grateful to various colleagues for useful comments on our earlier drafts of this paper. In particular we thank Philippe Delanoë, Gregoire Loeper and Cedric Villani, who are our partners in a French-Australian exchange programme. Around the same time, Kim and McCann [KM] found an alternative, (and more direct), approach to the full c-concavity of potentials, (indicated in Remark 3.5), for which we are also grateful. 2. Proof of Theorem Convexities relative to cost functions [MTW]. Let u be a semi-concave function in Ω, namely u C x 2 is concave for a large positive constant C. The supergradient + u [GM] and c-supergradient c + u are defined by (2.1) (2.2) + u(x 0 ) = {p R n : u(x) u(x 0 ) + p (x x 0 ) + o( x x 0 )}, + c u(x 0 ) = {y R n : u(x) c(x, y) c(x 0, y) + u(x 0 ) + o( x x 0 )} for x near x 0, where x 0 Ω. For a set E Ω, we denote + u(e) = x E + u(x) and c + u(e) = x E c + u(e). By (1.5) we have (2.3) c x (x 0, + c u(x 0 )) = + u(x 0 ). Note that + u(x 0 ) is a closed, convex set. Hence + c u(x 0) is closed and c-convex with respect to x 0. We may extend the above mappings to boundary points. Let x 0 Ω be a boundary point, we denote + u(x 0 ) = {p R n : p = lim k p k }, where p k + u(x k ) and {x k } is a sequence of interior points of Ω such that x k x 0, and let + c u(x 0) be given by (2.3). The c-normal mapping T u is defined by (2.4) T u (x 0 ) = {y R n : u(x) c(x, y) c(x 0, y) + u(x 0 ) for all x Ω}. Note that T u (x 0 ) + c u(x 0). c-support: Let u be a semi-concave function in Ω. A local c-support of u at x 0 Ω is a function of the form h = c(, y 0 ) + a 0, where a 0 is a constant and y 0 R n, such that u(x 0 ) = h(x 0 ) and u(x) h(x) near x 0. If u(x) h(x) for all x Ω, then h is a global c-support (or c-support for short) of u at x 0. If h is a local c-support of u at x 0, then y 0 + c u(x 0) and a 0 = u(x 0 ) c(x 0, y 0 ). c-concavity of functions: We say a semi-concave function u is locally c-concave if for any point x 0 Ω and any y + c u(x 0), h = c(, y) c(x 0, y) + u(x 0 ) is a local c-support of u at x 0. We say u is c-concave if for any point x 0 Ω, there exists a global c-support 4

5 at x 0 in Ω. We say u is strongly c-concave if it is both locally c-concave and c-concave. We say u is fully c-concave, (with respect to a target set V), if it is locally c-concave and every local c-support of u, (with T h V ), is a global c-support. We say u is strictly c-concave if it is fully c-concave and every c-support of u contacts its graph at one point only. c-segment: A set of points l R n is a c-segment with respect to a point y 0 R n if D y c(l, y 0 ) is a line segment in R n. c-convexity of domains: We say a set U is c-convex with respect to another set V if the image c y (U, y) is convex for each y V. Equivalently, U is c-convex with respect to V if for any two points x 0, x 1 U and any y V, the c-segment relative to y connecting x 0 and x 1 lies in U. By (2.3), a c-convex domain is topologically a ball. By definition, a c-concave function u can be represented as [GM] u(x) = inf{c(x, y) a(y) : y T u (Ω)}, where a is a function of y only. By (1.4), a potential function u is c-concave [GM]. If u is C 1, then the local c-support is unique and c-concavity is equivalent to full c-concavity. We also remark that a potential function may not be locally c-concave in general. Similarly we can define c -segment, c -support, c -convexity and c -concavity by exchanging variables x and y [MTW]. In this paper, we will generally omit the superscript when the meaning is clear Loeper s geometric property of (A3). Let y 0, y 1 be two points in Ω. Let ŷ 0 y 1 be the c-segment relative to a point x 0 Ω, connecting y 0 and y 1. By definition, (2.5) ŷ 0 y 1 = {y t : c x (x 0, y t ) = p t, t [0, 1]}, where p t = tp 1 + (1 t)p 0, and p 0 = c x (x 0, y 0 ), p 1 = c x (x 0, y 1 ). Let (2.6) h i (x) = c(x, y i ) a i, i = 0, 1, h t (x) = c(x, y t ) a t, where t (0, 1), a 0, a 1 and a t are constants such that h 0 (x 0 ) = h 1 (x 0 ) = h t (x 0 ). Suppose (A3) holds. Then for x x 0, near x 0, 0 < t < 1, we have the inequality (2.7) h t (x) > min{h 0 (x), h 1 (x)}, which is a crucial ingredient in the remaining analysis of this paper. Inequality (2.7) follows from (1.9). Indeed, by a rotation of axes, we assume that p 1 = p 0 + δe n, where δ is a positive constant and e n = (0,, 0, 1) is the unit vector in the x n -axis. By (1.9), (2.8) d 2 dt 2 c ij(x, y(x, p t ))ξ i ξ j c 0 δ 2 5

6 for any unit vector ξ orthogonal to e n, where y = y(x, p t ) is given in (A1). Now (2.7) follows from (2.8); (for details see [L], also (2.24) and 3.5 and 3.6) An analytic formulation of the c-convexity of domains. If Ω is c-convex with respect to Ω, by definition, c y (Ω, y) is convex for any y Ω. Suppose 0 Ω and locally Ω is given by (2.9) x n = ρ(x ) with Dρ(0) = 0 such that e n is the inner normal at 0, where x = (x 1,, x n 1 ). Then at (0, y) (for a fixed point y), (2.10) xi c y = c i,y + c n,y ρ xi, [ xi x j c y, γ = c ij,y, γ + c n,y, γ ρ xi x j ] 0, where γ is the inner normal of c y (Ω, y) at c y (0, y). We may write (2.10) explicitly, [c ij,yl γ l + c n,yl γ l ρ ij ] 0. Make the linear transformation Then we have ŷ k = a kl y l. (2.11) [c ij,ŷk a kl a 1 mlˆγ m + c n,ŷk a kl a 1 mlˆγ mρ ij ] 0. Let a ij = c i,j (0, y). Then c xi,ŷ j = δ ij and ˆγ = e n. We obtain (2.12) [c ij,ŷn + ρ xi x j ] 0, which is equivalent to (2.13) [c ij,yk c k,n + ρ xi x j ] 0. Let ϕ C 2 (Ω) be a defining function of Ω. That is ϕ = 0, ϕ 0 on Ω and ϕ < 0 in Ω. From (2.13), we obtain an analytic formulation of the c-convexity of Ω relative to Ω, which is independent of the choice of ϕ, [TW], (2.14) [ϕ ij (x) c k,l (x, y)c ij,k (x, y)ϕ l (x)] 0 x Ω, y Ω. Conversely, if Ω is connected and (2.14) holds, then Ω is c-convex. Following [TW], we call Ω uniformly c-convex with respect to Ω if the matrix in (2.13) or equivalently (2.14) is uniformly positive Geometric properties of the c-convexity of domains. Let y 0, y 1 be any two given points in Ω. Denote (2.15) N = N y0,y 1,a = {x R n : c(x, y 0 ) = c(x, y 1 ) + a}, 6

7 where a is a constant. Assume that the origin 0 N and locally N is represented as (2.16) x n = η(x ) such that η(0) = 0 and Dη(0) = 0 (obviously η also depends on y 0, y 1 and a). Then (2.17) c(x, η(x ), y 0 ) = c(x, η(x ), y 1 ) + a. Differentiating (2.17), we get (2.18) c i (0, y 0 ) + c n η i = c i (0, y 1 ) + c n η i, c ij (0, y 0 ) + c n η ij = c ij (0, y 1 ) + c n η ij. and hence we obtain (2.19) [c n (0, y 1 ) c n (0, y 0 )]η ij + [c ij (0, y 1 ) c ij (0, y 0 )] = 0. Now let y 0 be fixed but let y 1 and a vary in such a way that y 1 y 0 and the set N = N y0,y 1,a, given by (2.15), is tangential to {x n = 0}, namely η(0) = 0 and Dη(0) = 0. By a linear transformation as in 2.3 we assume that c xi,y j = δ ij at x = 0 and y = y 0. Then y 1 y 0 y 1 y 0 e n. We obtain (2.20) c n,n (0, y 0 )η 0 ij + c ij,n(0, y 0 ) = 0, where η 0 is the limit of η y0,y 1,a as y 1 and a vary as above. Now let Ω be c-convex, (uniformly c-convex), with respect to Ω. Suppose that Ω is given by (2.9) with Dρ(0) = 0 so that Ω is tangential to N at the origin. Then by (2.12)and (2.20) we obtain (2.21) D 2 (ρ η 0 ) 0, (> 0), at 0. From (2.21) we obtain some useful geometric properties of c-convex domains, assuming that c satisfies (A3). First, if Ω is c-convex with respect to Ω, then for any compact subset G Ω, Ω is uniformly c-convex with respect to G. Indeed, let y 0, y 1 be two points in Ω. Let p 0 = c x (0, y 0 ), p 1 = c x (0, y 1 ). For t (0, 1), let p t = tp 1 + (1 t)p 0 and y t satisfy c x (0, y t ) = p t. Then the set {p t : 0 t 1} is a line segment and the set {y t : 0 t 1} is a c-segment. Suppose p 1 = p 0 + δe n for some δ > 0. Let h t (x) = c(x, y t ) + a t, where a t = c(0, y 0 ) c(0, y t ) such that h t (0) = h 0 (0) for all t [0, 1]. Let N t = {x R n : h t (x) = h 0 (x)} and suppose that near 0, N t is given by x n = η t (x ). Since c x (0, y t ) = p t, we see that N t is tangential to {x n = 0}, namely D(η t η t )(0) = 0. By (2.8) we have furthermore the monotonicity formula (2.22) D 2 (η t η t )(0) > 0 7

8 for any t > t and t, t [0, 1]. Geometrically it implies that N t lies above N t if t > t, namely η t (x ) η t (x ) for x near 0, and equality holds only at x = 0. Consequently we obtain from (2.21) the strict inequality (2.23) D 2 (ρ η) > 0 at 0. From (2.23) we obtain the above mentioned property. Next, for any y 0, y 1 Ω, if N (given in (2.15)) is tangent to Ω at some point x 0 and if Ω is c-convex with respect to Ω, then the whole domain Ω lies on one side of N. Indeed, we may assume x 0 = 0, locally Ω and N are given respectively by (2.9) and (2.16), such that Dρ(0) = Dη(0) = 0. By (2.23), ρ(x) > η(x) for x near 0, x 0. Denote U = U y0,y 1,a = {x R n : c(x, y 0 ) < c(x, y 1 ) + a} so that N = U. If Ω does not lie on one side of N, namely Ω is not contained in U, then Ω U contains two disconnected components (one is the origin). Since Ω is a closed, compact hypersurface, we decrease the constant a (shrinking the set U) until a moment when two components of Ω U meet each other at some point x Ω. But since N is tangent to Ω at x, we reach a contradiction by (2.23). It follows that if Ω is c-convex with respect to Ω, then Ω = U y0,y 1,a, where the intersection is for all y 0, y 1 Ω and constant a such that U y0,y 1,a Ω. The above properties also extend to cost functions satisfying A3w and uniformly c- convex domains; (see [TW], Section 7). We remark that the monotonicity of N t also implies a weak form of (2.7), namely for x near x 0 and 0 < t < 1, (2.24) h t (x) min{h 0 (x), h 1 (x)}. Observe that a cost function satisfying (A3w) can be approximated locally by cost functions satisfying (A3), for example by subtracting ǫ x x 0 2 y y 0 2 for small ǫ. Hence the local monotonicity of N t is preserved under (A3w), whence (2.24) holds under (A3w) Local c-support is global. Let u be a locally c-concave function in Ω. Let h(x) = h a (x) = c(x, y 0 )+a be a local c-support of u at x and suppose Ω is c-convex with respect to y 0. Then h is a global c-support of u, namely (2.25) u(x) h(x) x Ω. Indeed, if this is not true, then for ε > 0 small, the set {x Ω : u(x) > h a ε (x)} contains at least two disconnected components. We increase ε (moving the graph of h vertically downwards) until at a moment ε = ε 0 > 0, two components first time touch each other at some point x 0. If x 0 is an interior point of Ω, by definition h a ε0 cannot be 8

9 a local c-support at x 0, which implies that y 0 + c u(x 0 ). We claim that for a sufficiently small r > 0, y 0 does not lie in the set + c u(b r(x 0 )) either. Indeed, if h k = c(, y 0 ) + a k for k = 1, 2, is a sequence of local c-support of u at x k and if x k, a k x 0, a 0, we have y 0 + c u(x 0) as u is semi-concave. Hence h 0 = c(, y 0 ) + a 0 is a local c-support of u at x 0. This is a contradiction. Therefore h a ε0 and u are transversal near x 0, and for ε < ε 0, close to ε 0, locally the set {x Ω : u(x) > h a ε (x)} cannot contain two disconnected components. Hence x 0 must be a boundary point of Ω. In case x 0 is a boundary point of Ω, we will also reach a contradiction by (2.23). Similarly as above, h a ε0 cannot be a local c-support at x 0, namely y 0 + c u(x 0). Without loss of generality let us assume that x 0 = 0 and locally Ω is tangent to {x n = 0} such that e n is an inner normal of Ω at 0. As before we also assume that c xi,y j = δ ij at x = 0 and y = y 0. Let p 0 = c x (0, y 0 ). By subtracting a linear function of x from both c and u, we assume that {x n = 0} is a tangent plane of h at 0. Then we have p 0 = 0, u(0) = 0, and u(x) o( x ) as x 0. Since y 0 + c u(x 0 ), we have 1 (2.26) β = lim t 0 t u(te n) < 0. Let p 1 = βe n, p t = tp 1 + (1 t)p 0 for t [0, 1], and let y t R n be determined by c x (0, y t ) = p t. Then p 1 + u(0) and y 1 c + u(0). Since u is locally c-concave, c(x, y 1 ) + b is a local c-support of u at 0, where b is a constant such that c(0, y 1 ) + b = u(0) = 0. Denote U = {x R n : h a ε0 (x) > c(x, y 1 ) + b} and N = U = {x R n : h a ε0 (x) = c(x, y 1 )+b}, such that 0 N. Since Ω is c-convex, by (2.23) we see that Ω U. But since c(x, y 1 )+b is a local c-support of u at 0, we have (2.27) h a ε0 (x) > c(x, y 1 ) + b u(x) for x Ω, near the origin. We reach a contradiction by our choice of ε 0. By approximation, we conclude (2.25) for arbitrary c-convex domains. From the above proof, we also see that if h is a c-support of u, then the contact set {x Ω : h(x) = u(x)} cannot contain two disconnected components (or points). In other words, the contact set is connected. We also remark that if u is a potential function to the optimal transportation problem with positive mass distributions f and g so that u is uniquely determined up to a constant, the boundary point case x 0 Ω can be reduced to the case x 0 Ω by extending u to larger domains, and it is not necessary to define the mapping + u and c + u on boundary points. Taking account of our remark at the end of the previous section, we also see that (2.25) also holds for (A3w) costs and uniformly c-convex domains. Consequently we obtain an alternate proof of the c-convexity of the solutions in Section 6 of [TW]. Furthermore by domain approximation, we may then extend (2.25) further to (A3w) costs and c-convex domains. The desired approximations follows by pulling back from an approximation of 9

10 the convex image c y (, y 0 )(Ω) by uniformly convex domains. Similarly we can extend our result in the previous section that Ω lies in one side of the set N to (A3w) costs and c-convex domains. We formulate the resultant conclusions as a theorem, assuming (also for the ensuing corollaries that (A1), (A2) and (A3w) are satisfied. Theorem 2.1. Let u be locally c-concave in Ω and suppose Ω is c-convex with respect to + c u(ω) Ω. Then u is fully c-concave in Ω and every contact set of u is connected. By local approximation by uniformly c-convex functions, we also infer the following condition for degenerate elliptic solutions of (1.6) to be c-concave. Corollary 2.2. Let u C 2 (Ω) and suppose Ω is c-convex with respect to Du(Ω) Ω. Then if (2.28) D 2 x c D2 u in Ω, u is c-concave in Ω. Similar to Corollary 2.2, we may obtain the full c-concavity of the functions introduced in 2.2. Corollary 2.3. Let h t, t [0, 1] be given by (2.6). Then for any s, t [0, 1], s < t, the function u s,t = min(h s, h t ) is fully c-concave in Ω. To prove Corollary 2.3, we locally approximate the functions h t by uniformly c-convex functions h t near points x 0 for which h s = h t. Then the mollifications of the functions, u s,t = min(h s, h t ) satisfy (2.28), (see [TW], Section 7), and hence u s,t is locally c-concave. By appropriate extensions of Ω, we conclude the full c-concavity of u s,t. Corollary 2.3 also follows directly from the proof of Theorem 2.1. Indeed, we observe that u s,t is fully c-concave if and only if N s,t = {x R n : h s (x) = h t (x)} is monotone in s and t, namely if s > s or t > t, then N s,t lies on one side of N s,t (same order for both s and t). For any fixed t, consider u s,t in a domain Ω t containing Ω, uniformly c-convex with respect to y t (see (2.5)). Then Ω t is uniformly c-convex with respect to y t for t near t. Hence u s,t is fully c-concave when t s small. It follows that N s,t is monotone. A third proof of Corollary 2.3 will be given in the next subsection, by considering the dual function of u s,t Potential functions. Let (u, v) be potential functions to the optimal transportation problem (1.2). Then for any point x 0 Ω, by (1.4), there exists a point y 0 Ω such that u(x 0 ) + v(y 0 ) = c(x 0, y 0 ). Hence is a c-support of u at x 0, and h(x) = c(x, y 0 ) v(y 0 ) (2.29) h (y) = c(x 0, y) u(x 0 ) 10

11 is a c-support of v at y 0. If u is C 1 at x 0, it has a unique c-support at x 0. As a potential function is semi-concave, it is twice differential a.e. in Ω. Hence u has a unique c-support almost everywhere. Next we consider the case when u is not C 1 at x 0. Let us first introduce the terminology extreme point. Let E be a convex set in R k. We say a point p E is an extreme point of E if there exists a plane P such that P E contains only the point p. It is easy to show, by induction on dimensions, that any interior point in a convex set can be expressed as a linear combination of extreme points. If u is not C 1 at an interior point x 0, since u is semi-concave, + u(x 0 ) is a convex set of dimension k for some integer 1 k n. Let N e u (x 0) denote the set of extreme points of + u(x 0 ). Let (2.30) T e u (x 0) = {y R n : c x (x 0, y) N e u (x 0)}. Then for any y 0 T e u (x 0), the function h(x) = c(x, y 0 ) + a is a global c-support of u at x 0, where a is a constant such that h(x 0 ) = u(x 0 ). This assertion follows from a similar one for concave functions, which can be proved by blowing up the graph of u to a concave cone. That is for any p T e u (x 0), there exists a sequence of C 1 -smooth points {x k } of u, x k x 0, such that the c-support of u at x k converges to a c-support of u at x 0. Recall that at C 1 points, u has a unique global c-support. The reader is referred to [V] for further amplification. From the above assertion and (2.7) it follows that a potential function is a local c- concave function if (A3) is satisfied. That is for any y + c u(x 0 ), the function h(x) = c(x, y ) c(x 0, y ) + u(x 0 ) is a local c-support of u at x 0. By (1.4), h is a global c-support if (2.31) u(x 0 ) + v(y ) = c(x 0, y ). We remark that in general a potential function may fail to be locally c-concave (with respect to the definition in 2.1), if (A3w) is violated. From the above assertion it also follows that if u is not C 1 at x 0, then the function h in (2.29) is a c-support of v at any point y 0 T e u (x 0). In other words, T e u (x 0) is contained in the contact set (2.32) C = {y Ω : h (y) = v(y)}. By (2.31), a local c-support of u at x 0 is a global one if and only if the contact set C is c-convex (with respect to x 0 ). However, in 2.5 we proved that C is connected if Ω is c-convex. By applying this result directly to the c-transform u 01 of the function u 01 = 11

12 min{h 0, h 1 }, where h 0, h 1 are given in (2.6) and y 0, y 1 T u (x 0 ), we obtain + c u(x 0 ) = T u (x 0 ). For this we observe that T u01 (x 0 ) T u (x 0 ) ŷ 0 y 1, whence ŷ 0 y 1 = T u01 (x 0 ) T u (x 0 ), (see also [L], Proposition 1.12). Consequently, we conclude that the potential u is fully c-concave. Note that this also follows from Corollary 2.3. When f and g are positive, we may alternatively extend the potential u to larger domains, to conclude its full c-concavity from its local c-concavity. Since Ω is c-convex with respect to Ω, from the argument in 2.5, we also have that any local c-support h of v, with c-normal image x 0 Ω, is a global one. Hence v is fully c-concave in Ω, with respect to Ω. Taking account of our remark at the end of 2.4, we see that we also obtain, by this approach, the full c-concavity of u and v under (A3w), (as in Corollary 2.3). Furthermore, u is C 1 smooth if and only if its dual function v is strictly c-concave. 2.7 Proof of Theorem 1.1. For the proof of Theorem 1.1, we employ the technique of Perron lifting. Let (u, v) be the potential functions to (1.2). To prove Theorem 1.1, it suffices to prove that v is strictly c-concave. By approximation we may assume that f, g are positive and smooth. By 2.4 we may also assume that Ω is uniformly c-convex with respect to Ω. From 2.6, v is fully c-concave with respect to Ω so that every local c-support of v, (with c-normal image in Ω), is a global one. Suppose to the contrary that v is not strictly c-concave. Then there is a c-support h of v at some point y 0 Ω such that the contact set C = {y Ω : h (y) = v(y)} contains more than one point. From the argument in 2.5, C is connected. Hence for any r > 0 small, the intersection B r (y 0 ) C is not empty. Let {B j : j = 1,, k} be finitely many balls with radius r/2, centered on B r (y 0 ), such that k j=1 B j B r (y 0 ). Denote v 0 = v. For j = 1,, k, let v j C 3 (B j ) C 0,1 (B j ) be the elliptic solution of (2.34) det(d 2 yc D 2 w) = 1 2 δ 0 in B j, w = v j 1 on B j, where (2.35) δ 0 = inf detc i,j (g/f), is positive by the hypotheses of Theorem 1.1 and v j is extended to all of Ω such that v j = v j 1 in Ω B j. The solvability of the problems (2.34) is proved similarly to the corresponding problem in [MTW], Section 5, by first approximating the boundary data and solving for globally smooth solutions utilizing the existence of appropriate barriers for 12

13 sufficiently small r. From the interior second derivative estimates in [MTW], (Theorem 4.1) and global gradient estimates, ( resulting from [MTW], Lemma 5.2), we infer the solvability of (2.34) in C 3 (B j ) C 0,1 (B j ). From our construction, the functions v j will be locally c-concave in Ω. In particular v =: v k is locally c-concave in Ω and from 2.5, it is fully c-concave, with respect to Ω. By the comparison principle, (as used in [MTW], Lemma 4.2), we have v 1 < v 0 in B 1 and by induction, v j < v 0 in B j for all j = 2,, k. It follows that v < v near B r (y 0 ). We have therefore obtained another fully c-concave function v which satisfies v = v in B r/2 (y 0 ) {Ω B 3r/2 (y 0 )}, v < v near B r (y 0 ). Hence the contact set {y Ω : h (y) = v(y)} cannot be connected. But this is impossible from the argument in 2.5 (as remarked at the end of 2.5). Hence v is strictly c-concave. This completes the proof of Theorem Remarks 3.1. The full c-concavity of the potential u, as proved in 2.5 and 2.6, completes the proof of the regularity results in [MTW], (and consequently [L, v1]) in their full generality. In [MTW] we introduced a notion of generalized solution to the boundary value problem (1.6) and proved interior regularity under the conditions in Theorem 1.1, assuming also the smoothness of f and g. If the potential function u is not fully c- concave, a local c-support of u may not be a global one. In such case, the definition of generalized solution in [MTW] is not local and the (two-sided) comparison principle may not hold in arbitrary sub-domains. Theorem 1.1 rules out the possibility provided the cost function satisfies (A1)-(A3) and Ω is c-convex with respect to Ω, as assumed in [MTW]. More specifically the full c-concavity of u justifies the implication w u in the uniqueness argument at the end of the proof of Theorem 2.1 in [MTW] Note that the full c-concavity of u was not used to complete the proof of Theorem 1.1 in 2.7 and follows from the C 1 regularity by approximation. This was our approach in the original draft of this paper and we are grateful to Gregoire Loeper for suggesting that it was already proved in the preceding sections, (by virtue of the equivalence between the connectivity and the c-convexity of the contact set C ). In fact we have added three further ways of deducing the full c-concavity of u from our earlier analysis.we may also use c-convexity, rather than connectivity, in the proof of Theorem 1.1 by solving just one Dirichlet problem to reach our contradiction. Note also that in 2.7, we also only needed the comparison argument on its good side, so the full c-concavity of v can also be dispensed with there In certain cases the full c-concavity is a direct consequence of the geometric form (2.7) of condition (A3). In particular we have in two dimensions, 13

14 Corollary 3.1. Suppose the cost function c satisfies (A1)-(A3). Then any potential functions (u, v) defined in the whole R 2 are fully c-concave. Indeed, for any point x 0 Ω, if u is C 1 at x 0, there is a unique global c-support of u at x 0. Otherwise, let y 0, y 1 be any two points in + c u(x 0). Let h 0 (x) = c(x, y 0 ) a 0 and h 1 (x) = c(x, y 1 ) a 1 be two c-supports of u at x 0. Denote N = {x R n : h 0 (x) = h 1 (x)}. N is a curve which divides R 2 into two parts, and both are non-compact. It suffices to show that (3.1) h t h 0 (= h 1 ) on N, where t (0, 1) and h t is as in (2.7). But if (3.1) is not true, by moving the graph of h t downwards, we see that there is a constant b and a point x N such that (3.2) (3.3) h t (x ) b = min{h 0 (x ), h 1 (x )}, h t (x) b min{h 0 (x), h 1 (x)} x N near x. But this is in contradiction with (2.7) at x. Hence Corollary 3.1 holds. Note that for potential functions on bounded domains, by the uniqueness of potential functions when restricted to {f > 0}, we see that u (and similarly v) is fully c-concave when restricted to {f > 0}. Corollary 3.1 also extends to (A3w) costs, (see 3.6) The proof of Corollary 3.1 does not extend to higher dimensions, as we don t know if there is a point x N such that (3.2) and (3.3) hold. But Corollary 3.1 holds on compact manifolds of any dimension, as the set N is compact. In particular it holds for the reflector design problem (in the far field case) [W1, W2]. But for the reflector design problem, a c-support is a paraboloid with focus at the origin and one can also verify (3.1) directly [CGH]. We also note here that the full c-concavity and consequent C 1 regularity is also included in [L], (by approximation from [GW]) We remark that if the cost function c does not satisfy (A3), then (2.7) may not hold. Loeper [L] shows that if condition (A3w) is violated, there are potential functions which are not fully c-concave. Furthermore, the potential function u may not be C 1 smooth even if both f and g are positive and smooth. Note that Loeper s potential functions are the negative of those here so that our c-concavity is equivalent to his c- convexity By approximation from our global regularity [TW], Loeper [L] proved the full c-concavity of the potential u under the weaker hypothesis (A3w) but assuming that the domains Ω and Ω lie in mutually uniformly c-convex domains. After this paper was originally written, we learnt from Robert McCann that he and Y.Kim, (see [KM]), had independently found a direct proof with the containing domains only needed to be c-convex. Their proof is based on direct calculation that under (A3), or (A3w), (3.4) d 2 dt 2h t(x) ( c 0 /2) ξ 2 η 2, 14

15 for d dt h t(x) = ξ.η = 0, where ξ = c y (x, y t ) c y (x 0, y t ), η = c x,y (x 0, y t )(p 1 p 0 ), from which it follows that the contact set of the potential function u with its c-support is connected. Here h t is the function given in (2.6) and we observe (3.4) follows directly from (1.9), (with x and y interchanged). Note that (3.4) implies directly that Loeper s geometric property (2.7) holds for all x Ω {0}, thereby providing a simple proof of the full c-concavity of u in 2.6, independent of the preceding sections For completeness, we also indicate a short proof of (2.7), (extended to (A3w)), along the same lines as in [L]. With the notation of 2.2, we assume that h t (x 1 ) min{h 0 (x 1 ), h 1 (x 1 )} for some x 1 Ω. Then from Taylor s formula and (2.8), we infer the same inequality for (x x 1 ).ξ = 0 and x near x 1. Since (2.7) is clearly true for x 1 x 0 = te n for sufficiently small t > 0, we are done Finally we remark that we may weaken our notion of local c-concavity, by saying that the function u is weakly locally c-concave in the domain Ω, if there is a local c- support at each point x 0 Ω. By combining our arguments in 2.5 and 2.6, we obtain a further corollary. Corollary 3.2. Let u be weakly locally c-concave in Ω, where Ω is c-convex, with respect to c + u(ω) and c satisfies (A1),(A2) and (A3w). Then u is fully c-concave. References [C1] Caffarelli, L.A., A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. Math., 131(1990), [C2] Caffarelli, L.A., The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5(1992), [C3] Caffarelli, L.A., Boundary regularity of maps with convex potentials II. Ann. of Math. 144 (1996), no. 3, [C4] Caffarelli, L.A., Allocation maps with general cost functions, in Partial Differential Equations and Applications, (P. Marcellini, G. Talenti, and E. Vesintini eds), Lecture Notes in Pure and Appl. Math., 177(1996), pp [CGH] Caffarelli, L.A., Gutierrez, C. and Huang, Q., On the regularity of reflector antennas, Ann. of Math., to appear. [D] Delanoë, Ph., Classical solvability in dimension two of the second boundary value problem associated with the Monge-Ampère operator, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 8(1991), [GM1] Gangbo, W. and McCann, R.J., Optimal maps in Monge s mass transport problem, C.R. Acad. Sci. Paris, Series I, Math. 321(1995), [GM2] Gangbo, W. and McCann, R.J., The geometry of optimal transportation, Acta Math., 177(1996), [GT] Gilbarg, D., Trudinger, N.S., Elliptic partial differential equations of second order, Springer, [GW] Guan, P.F. and Wang, X-J., On a Monge-Ampère equation arising in geometric optics, J. Diff. Geom., 48(1998), [GM2] Kim, Y-H. and McCann, R.J., On the cost-subdifferentials of cost-convex functions, arxiv:math/ [L] Loeper, G., Continuity of maps solutions of optimal transportation problems, arxiv:math/ [MTW] Ma, X.N., Trudinger, N.S. and Wang, X-J., Regularity of potential functions of the optimal trans- 15

16 portation problem, Arch. Rat. Mech. Anal., 177(2005), [TW] Trudinger, N.S. and Wang, X-J., On the second boundary value problem for Monge-Ampère type equations and optimal transportation, arxiv:math/ [U1] Urbas, J., On the second boundary value problem for equations of Monge-Ampère type, J. Reine Angew. Math., 487(1997), [U2] Urbas, J., Mass transfer problems, Lecture Notes, Univ. of Bonn, [V] Villani, C., Topics in optimal transportation problem, Amer. Math. Soc., [W1] Wang, X.J., On the design of a reflector antenna, Inverse Problems, 12(1996), [W2] Wang, X.J., On the design of a reflector antenna II, Calc. Var. PDE, 20(2004), Neil S. Trudinger, Centre for Mathematics and Its Applications, Australian National University, Canberra, ACT 0200, Australia address: neil.trudinger@anu.edu.au Xu-Jia Wang, Centre for Mathematics and its Applications, Australian National University, Canberra ACT 0200, Australia address: wang@maths.anu.edu.au 16