A VARIATIONAL METHOD FOR THE ANALYSIS OF A MONOTONE SCHEME FOR THE MONGE-AMPÈRE EQUATION 1. INTRODUCTION

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1 A VARIATIONAL METHOD FOR THE ANALYSIS OF A MONOTONE SCHEME FOR THE MONGE-AMPÈRE EQUATION GERARD AWANOU AND LEOPOLD MATAMBA MESSI ABSTRACT. We give a proof of existence of a solution to the discrete problem obtained from a discretization of the Monge-Ampère equation by a monotone scheme. We prove that the unique minimizer of a convex functional of the gradient under convexity and nonlinear constraints solves the finite difference equation. The variational method with the monotone scheme gives a numerical method which is shown to be numerically stable when the right hand side of the equation is a linear combination of Dirac masses, i.e. for Aleksandrov solutions. 1. INTRODUCTION We are interested in the numerical resolution of the Monge-Ampère equation (1.1) det 2 u = f in Ω, u = g on Ω, where the domain Ω R n is assumed to be bounded and convex with boundary Ω. For a smooth function u, 2 u is the Hessian of u with entries ( 2 u)/( x i x j ), i, j = 1,..., n. In general the expression det 2 u should be interpreted in the sense of Aleksandrov, c.f. Section 4. The functions f and g are given with f 0. We make the assumption that f L 1 (Ω) and g C 0,1 ( Ω) can be extended to a function g C(Ω) which is convex on Ω. Under these assumptions, (1.1) is known to have a unique convex Aleksandrov solution [11]. To discretize (1.1) one needs a notion of discrete convexity as well as a discretization of det 2 u. As explained in [4, Section 1.2], it has been an open problem to prove the existence of a solution to the discrete problem obtained when one uses the monotone scheme of [7] and a notion of discrete convexity, also used in [7], which we will refer to as wide stencil convexity. Our strategy is based on a remark of P.L. Lions [13] which asserts that the unique C 0,1 (Ω) solution of (1.1) in the sense of Aleksandrov is the unique minimizer of J(w) = Φ( w) dx, over the convex set (1.2) Ω S = { v C(Ω), v = g on Ω, v convex on Ω, (det 2 v) 1 n f 1 n in Ω }, where Φ is a nonnegative strictly convex function satisfying (2.1) below. Mimicking the above principle at the discrete level, we prove the existence of a unique minimizer to the discrete optimization problem and shows that the unique minimizer solves the finite difference equations obtained with the monotone discretization. The notation C 0,1 (Ω) stands for the set of Lipschitz continuous functions on Ω. We present numerical evidence of stability of the discretization with a test case for which the right hand side is a linear combination of Dirac masses. We do not address in this paper the convergence of the discretization proposed in [7] to the Aleksandrov solution and refer to [5] for an axiomatic This research was supported in part by NSF grant DMS to Gerard Awanou and NSF grant DMS to the Mathematical Biosciences Institute. 1

2 2 GERARD AWANOU AND LEOPOLD MATAMBA MESSI approach to the convergence of finite difference schemes to the Aleksandrov solution. The existence of a solution to the discrete problem is proved in this paper, while the other assumptions required in [5], except stability, are proved in [2]. Organization of the paper. The paper is organized as follows: in the next section, we introduce some notation and prove that when the equation is discretized with the monotone scheme of [7], the unique minimizer of the discrete problem solves the finite difference equation. Thus the latter has at least one solution. Section 3 is devoted to numerical results. In Section 4 we review the notion of Aleksandrov solution and explicitly compute the Monge-Ampère measure for a test function often used to evaluate the efficiency of numerical methods for Aleksandrov solutions [6]. 2. DISCRETE OPTIMIZATION PROBLEM We start with some notations needed to state the discrete optimization problem. All the functions we consider take finite values. We make the usual convention of denoting constants by the letter C. We make the following assumptions on the nonnegative convex function Φ (2.1a) Φ C 2 (R n ) (2.1b) Φ(p) C + C p (2.1c) ν > 0 such that Φ(p) Φ(q), p q ν p q 2, p, q R n where, denotes the Euclidean inner product in R n and. denotes the associated norm. Without loss of generality, we will assume, for the analysis in this paper, that Φ is strictly convex. For example, one may take Φ(p) = p 2. The results in [13] hold for more general convex functions Φ Notation and preliminaries. Most of the notation is taken from [1]. Let Ω = (0, 1) n, h = 1/N, N N and let M h denote the mesh which consists of points x = hz R n Ω, z Z n. Put M h = M h Ω, M h = M h Ω, and denote by U h the set of real valued functions defined on M h. Without loss of generality, for a function v defined on Ω, we use the notation v for the restriction of v to M h. We denote by e i, i = 1,..., n the i-th coordinate unit vector of R n and consider the first order difference operator defined on M h by v i h (x) := v h(x) v h (x he i ). h 2.2. Discretization of the functional J. We recall that for a convex function v on Ω, the one-sided partial derivatives are defined everywhere and denote by v the left-hand side gradient of v. Since the convex function v is differentiable almost everywhere, we have (2.2) J(w) = Φ( w) dx. Ω We now use the latter expression of J(w) to construct a discrete analogue of J(w) using Riemann sums. For x = (x 1,..., x n ) M h, we define P x = { y Ω, x i h y i x i }. It is easy to see that P x is nonempty if and only if either dist(x, Ω) < h n or dist(x h, Ω) < h n. Furthermore, we have x Mh P x = Ω and for x y, P x P y is a set of Lebesgue measure 0. For x M h, we denote by P x the Lebesgue measure of P x, i.e. P x = h n if P x Ω. We

3 use the notation x h/2 to denote the center of P x that is, the point obtained by subtracting h/2 from each coordinate x i of x. We define for v h U h and a convex real-valued function Φ on R n J h (v h ) = x M h h n Φ( h v h (x)), where h v h is the vector of backward finite differences of the mesh function v h, i.e. h v h (x) = ( i v h (x)),...,n. We assume that the sum in the definition of J h (v h ) is over the set of mesh points at which h v h (x) is defined. We note that for all i, i v h extends to a piecewise constant function on Ω, denoted also by i v h and taking the constant value i v h (x) on P x for x M h Monotone discretization of det 2 u and wide stencil convexity. We prove in this section that if one uses the monotone scheme introduced in [7], the discrete optimization problem has a unique minimizer which solves the corresponding finite difference equations. This also proves the existence of a solution to the latter. We do not know whether uniqueness holds for a finite difference system derived from a monotone scheme approximation of the Monge-Ampère operator. However the variational approach in this paper provides a method for selecting a specific solution of such a finite difference system. We define dθ(x) = max min v αh v =1 α h /x±α h M h α h. and let ζ 0 M h such that dθ(ζ 0 ) = min dθ(x). x M h The quantity dθ(ζ 0 ) has been referred to as directional resolution in [7]. Following [7, 3], we define a monotone Monge-Ampère operator by n v h (x + α i ) 2v h (x) + v h (x α i ) (2.3) M[v h ](x) = inf α i 2 for x M h, (α 1,...,α n) W h where W h denotes the set of orthogonal bases of R n such that for (α 1,..., α n ) W h, ζ 0 ± α i M h, i. For x M h, such that x + α i M h but x α i / M h for some i, we denote by x αi the intersection with Ω of the line through x and x + α i closest to x. We then define v h (x α i ) to be the value obtained by quadratic interpolation of v h (x αi ), v h (x) and v h (x + α i ). Similarly, if x α i M h but x + α i / M h for some i, we define v h (x + α i ) by quadratic interpolation of v h (x αi ), v h (x) and v h (x α i ), where now x αi is the intersection with Ω of the line through x and x α i closest to x. If both x α i and x + α i are not in M h, we use as interpolation points x and the intersection points with the boundary of the line through x α i and x + α i. For x M h and e Zn such that e is an element of an orthogonal base (α 1,..., α n ) W h, we define e v h (x) = v h (x + e) 2v h (x) + v h (x e) with the above convention. Definition 2.1. We say that a mesh function v h is wide stencil convex if and only if e v h (x) = v h (x + e) 2v h (x) + v h (x e) 0 for all x M h and e Zn for which e v h (x) is defined. Let C h denote the cone of wide stencil convex mesh functions. We recall that the discrete Laplacian takes the form d h v h (x) = hei v h (x), 3

4 4 GERARD AWANOU AND LEOPOLD MATAMBA MESSI where { e i, i = 1,..., d } denotes the canonical basis of R d. We define (2.4) S h = { v h C h, v h = g on M h, and (M[v h ](x)) 1 n f(x) 1 n, x M h }. We seek a minimizer of J h over S h and consider the problem: find u h C h such that (2.5) M[u h ](x) = f(x), x M h u h = g on M h. Lemma 2.2. The operator v h M[v h ] is concave and thus the set S h is convex. Proof. We note that for λ 0 (M[λv h ](x)) 1 n = λ(m[vh ](x)) 1 n. It is therefore enough to prove that for v h, w h C h, we have (2.6) (M[v h + w h ](x)) 1 n (M[vh ](x)) 1 n + (M[wh ](x)) 1 n. Let (α 1,..., α n ) W h (x) be such that with M[v h + w h ](x) = n (λ i 1 + λ i 2) λ i 1 = v h(x + α i ) 2v h (x) + v h (x α i ) α i 2 and λ i 2 = w h(x + α i ) 2w h (x) + w h (x α i ) α i 2. Since v h, w h C h, λ i 1, λi 2 0 for all i. By Minkowski s determinant theorem ( n ) 1 ( (λ i 1 + λ i n n ) 1 ( n n ) 1 n 2) +, from which (2.6) follows. Lemma 2.3. The convex set S h is nonempty. λ i 1 λ i 2 Proof. For each y in M h, let q y be a quadratic function such that M[q y ](x) = f(y) for all x in M h. For example, we may define q y by q y (x) = 1 2 f(y) x 2. Put ˆf = y M q y. Since g is bounded on Ω, we can find a number K such that ˆf K g on h Ω. We define w h U h by We claim that w h S h. w h (x) = ˆf(x) K, x M h w h = g on M h. For x M h and (α 1,..., α n ) W h such that x ± α i M h for all i, either w h (x + α i ) = ˆf(x+α i ) K or w h (x+α i ) = g(x+α i ) ˆf(x+α i ) K. Similarly w h (x α i ) ˆf(x α i ) K. We conclude that w h (x + α i ) 2w h (x) + w h (x α i ) ˆf(x + α i ) 2 ˆf(x) + ˆf(x α i ) α i 2 f(x). The case when w h (x + α i ) or w h (x α i ) is obtained by quadratic interpolation is similar. The quadratic function passing through the points (x α i, w h (x α i )), (x, w h (x)) and (x αi, g(x αi )) is above the corresponding section of the graph of ˆf(x) K and thus w h (x + α i ) ˆf(x + α i ) K. A similar argument gives w h (x α i ) ˆf(x α i ) K and we conclude as above.

5 5 Thus w h C h and (M[w h ]) 1/n f. We can now state the following result. Theorem 2.4. For Φ(p) = p 2, the functional J h has a unique minimizer u h in S h and u h solves the finite difference equation (2.5). Proof. Since Φ is strictly convex and S h is nonempty, it follows that the functional J h has a unique minimizer u h on the convex set S h. We now show that u h solves the finite difference system (2.5). To this end, it suffices to show that M[u h ] = f h on M h. Let us assume to the contrary that there exists x 0 M h such that M[u h ](x 0 ) > f h (x 0 ) 0. We have (2.7) 0 f h (x 0 ) < M[u h ](x 0 ) n u h (x 0 + α i ) 2u h (x 0 ) + u h (x 0 α i ) α i 2, for all (α 1,..., α n ) W h. Using the geometric mean inequality, we get f h (x 0 ) 1 n < (M[uh ](x 0 )) 1 1 αi u h (x 0 ) n n α i 2. We conclude that for all e Z n such that e u h (x 0 ) is defined, we have e u h (x 0 ) > 0. Let ɛ 0 = inf{ e u h (x 0 ), e Z n }. The mapping E : r n (u h(x 0 + α i ) + u h (x 0 α i ) 2r)/ α i 2 is continuous and by (2.7), with r 0 = u h (x 0 ), E(r 0 ) > f h (x 0 ). Therefore there exists ɛ 1 > 0 such that for r r 0 < ɛ 1, we have E(r) > f h (x 0 ). Finally, put ɛ = min(ɛ 0, ɛ 1 ). Following a strategy used in [12], we define w h by w h (x) = u h (x), x x 0, w h (x 0 ) = u h (x 0 ) + ɛ 4. By construction w h = g h on M h. For x x 0 either e w h (x) = e u h (x) or e w h (x) = e u h (x) + ɛ/4. Moreover e w h (x 0 ) = e u h (x 0 ) ɛ/2 ɛ 0 ɛ/2 ɛ/2 > 0 by the definition of ɛ. We conclude that w h C h. For x x 0, M[w h ](x) M[u h ](x) f h (x). We have n αi w h (x 0 ) n M[w h ](x 0 ) = α i 2 = αi u h (x 0 ) ɛ 2 α i 2, for some element (α 1,..., α n ) W h. That is, M[w h ](x 0 ) = E(r 0 + ɛ/4) > f h (x 0 ). Thus M[w h ](x) f h (x) for all x M h.

6 6 GERARD AWANOU AND LEOPOLD MATAMBA MESSI It remains to show that J h (w h ) < J h (u h ). Let M x0 denote the subset of M h consisting in x 0 and the points x 0 + he j, j = 1,..., n at which h u h is defined. We have J h (w h ) = h n h u h (x) 2 + h n h w h (x 0 ) 2 + h w h (x 0 + he j ) 2 x/ M x0 j=1 J h (w h ) = h n h u h (x) 2 + h n 2 (w h (x 0 ) w h (x 0 he i )) 2 x/ M x0 + h n 2 (w h (x 0 + he j ) w h (x 0 + he j he i )) 2 = h n j=1 x/ M x0 h u h (x) 2 + h n 2 j=1 (w h (x 0 + he j ) w h (x 0 + he j he i )) 2 i j + h n 2 (w h (x 0 ) w h (x 0 he i )) 2 + (w h (x 0 + he i ) w h (x 0 )) 2. However (w h (x 0 ) w h (x 0 he i )) 2 + (w h (x 0 + he i ) w h (x 0 )) 2 = ( uh (x 0 ) u h (x 0 he i ) + ɛ ) 2 ( + uh (x 0 + he i ) u h (x 0 ) ɛ ) 2 = 4 4 Thus, by our choice of ɛ, (u h (x 0 ) u h (x 0 he i )) 2 + (u h (x 0 + he i ) u h (x 0 )) 2 + nɛ2 8 ɛ 2 hu h (x 0 ). J h (w h ) = J h (u h ) + nɛ2 8 ɛ 2 hu h (x 0 ) = J h (u h ) + ɛ 2 (nɛ 4 hu h (x 0 )) < J h (u h ), since e u h (x 0 ) ɛ 0 and thus h u h (x 0 )) nɛ 0 nɛ > nɛ/4. This contradicts the assumption that u h is a minimizer and concludes the proof. 3. NUMERICAL EXPERIMENTS In this section, we report numerical results for the variational framework studied in this paper. A natural approach for solving the nonlinear finite difference equations is to use Newton s method [7]. However, it has been reported in [6] that the latter is not effective for Test 6 below for which the right hand side is a linear combination of Dirac masses. To illustrate the numerical stability of the variational method studied, we present results for a wide range of classical test functions. Our numerical results indicate that the discretization proposed in [7] is stable for Aleksandrov solutions. This observation combined with the numerical results given in [5] indicate that it may be possible to solve the nonlinear finite difference equations effectively for Aleksandrov solutions. We refer to [6] for a convergent discretization for such solutions. Throughout this section, Ω denotes the square (0, 1) (0, 1) and we take Φ(p) = p 2.

7 3.1. The boundary value problems. We tested the framework developed in this paper on six examples of varying regularity and covering the full range of solutions of the Monge-Ampère equation Approximating classical and viscosity solutions. The first boundary value problem, admitting a classical smooth solution, is given by (3.1) f(x) = (1 + x 2 ) exp( x 2 ) and g(x) = exp( x 2 2 ). The second boundary value problem with data 2 (3.2) f(x) = (2 x 2 ) 2 and g(x) = 2 x 2 admits a viscosity solution with unbounded gradient. No exact solution is known for the third example with data (3.3) f(x) = 1 and g(x) = Approximating Aleksandrov solutions. The fourth example corresponds to the Monge-Ampère Dirichlet boundary value problem with data (3.4) f(x) = 0 and g(x) = x For the fifth example the right hand side is taken as a Dirac mass δ (1/2,1/2) which is approximated with the mesh functions { π if max{ x f h (x) = 4h , x } < h 0 otherwise. In that case the exact solution is given by u(x 1, x 2 ) = (x 1 1/2) 2 + (x 2 1/2) 2. For the last and sixth example, we solve the equation det 2 u = µ where the measure µ is given by (3.5) µ = π 2 δ ( 1 4, 1 2 ) + π 2 δ ( 3 4, 1 2 ). We show in the appendix that a solution to det 2 u = µ is the function x 2 { 1 2 if 1 4 < x 1 < 3 } 4 u(x) = min (x )2 + (x )2, (x 1 34 )2 + (x 2 12 )2 otherwise. The corresponding mesh functions approximating µ were taken as f h (x) = π/(8h 2 )1 Dh (1/4,1/2) + π/(8h 2 )1 Dh (3/4,1/2) with D r (x) denoting the open ball of center x and radius r in the maximum norm. For this test we experimented with various approximations of the Dirac masses with different levels of accuracy. In particular taking the closed ball in the definition of the above mesh function leads to divergence for the 17 point scheme Convex program. The convex program associated with the monotone scheme reads as follows arg min h 2 Φ( h u h (x)) x M h (3.6) subject to u h = g on M h M[uh ](x) f(x) x M h. 7

8 8 GERARD AWANOU AND LEOPOLD MATAMBA MESSI For our purposes in this section, we use the 9-point monotone scheme for which there are exactly two pairs of orthogonal directions for each interior grid point and the 17 point scheme with quadratic interpolation for which there are exactly six pairs of orthogonal directions for each interior grid point Implementation. It is not our goal in this section to develop or identify the most efficient method for solving (3.6). A plethora of methods and algorithms to solve this type of convex programs, including primal-dual and barrier methods, are readily available in the literature. Rather, we aim to leverage these algorithms to solve (3.6) and provide numerical evidence supporting the analysis done in this work. In particular, we use the convex optimization toolbox CVX [8, 9] to formulate the optimization problem in Matlab and solve it using the infeasible primal-dual interior point algorithm SDPT3 [14] Numerical evidence of convergence. We provide numerical evidence for the six boundary value problems defined early in this section. For Tests 3, 4, 5 and 6 we graph the numerical solutions in Figure 1. For Tests 1, 2, 5 and 6 we give the maximum errors using the 9 point scheme, Table 1 and the 17 point scheme Table 2. h Test 1 Test 2 Test 5 Test TABLE 1. Maximum error for the 9 point scheme. h Test 1 Test 2 Test 5 Test TABLE 2. Maximum error for the 17 point scheme. We note that the numerical solution is bounded independently of h. However for the two Dirac mass example, accuracy deteriorates with the 17 point scheme. The numerical error does decrease with h in that case. After all the numerical solution is an approximate minimizer and with more complicated constraints it seems more difficile to find a suitable discrete minimizer. For example the computing resources for solving the optimization problem for h = 2 6 with the 17 point scheme far exceed the ones required for the 9 point scheme although both problems have the same number of constraints. To further illustrate our point that the discretization considered is stable for Aleksandrov solutions, we repeat the numerical experiment for Test 6 with the standard finite difference discretization. It is a 9 point scheme which, for smooth solutions, is more accurate than the 9 point monotone scheme. One could have expected divergence for singular solutions. But we obtained accurate results for all

9 9 (A) Numerical solution for Test 3. (B) Numerical solution for Test 4. (C) Numerical solution for Test 5. (D) Numerical solution for Test 6. FIGURE 1. Solutions computed on a grid with the 9 point scheme. h 1/2 2 1/2 3 1/2 4 1/2 5 1/ TABLE 3. Maximum error for Dirac masses with standard discretization of the determinant of the Hessian. the test cases. Here we present the results for Test 6, c.f. Table 3, where we take as the right hand side f h (x) = 2/h 2 1 Bh/2 (1/4,1/2) + 2/h 2 1 Bh/2 (3/4,1/2) with B r (x) denoting the Euclidean open ball of center x and radius r. The inequality constraints are now taken as det(h[uh ](x)) f(x) for x M h, where H[u h ](x) is the matrix obtained by discretizing the Hessian with second order accurate central finite difference approximations. The code used to generate the results is available upon request. 4. APPENDIX In this section we review the geometric definition of the notion of Monge-Ampère measure and the notion of Aleksandrov solution. We refer to [10] for further details.

10 10 GERARD AWANOU AND LEOPOLD MATAMBA MESSI The normal mapping or subdifferential of a real valued function v defined on Ω, is a set-valued mapping v defined from Ω to the set of subsets of R d such that for any x 0 Ω, For a subset E Ω, we define v(x 0 ) = { q R n : v(x) v(x 0 ) + q (x x 0 ), for all x Ω }. v(e) = v(x). We denote by E the Lebesgue measure of E when E is measurable. The class is a Borel σ-algebra and the mapping x E S = { E Ω, v(e) is Lebesgue measurable }, M[v] : S R, M[v](E) = v(e), is a measure, finite on compact sets, called the Monge-Ampère measure associated with the function v. Definition 4.1. Let µ be a Borel measure on Ω. A convex function on Ω is an Aleksandrov solution of det 2 v = µ if M[v] = µ. We now show that the Monge-Ampère measure on Ω = (0, 1) (0, 1) associated with the function x 2 { 1 2 if 1 4 < x 1 < 3 } 4 u(x 1, x 2 ) = min (x )2 + (x )2, (x 1 34 )2 + (x 2 12 )2 otherwise, is given by M[u] = π 2 δ ( 1 4, 1 2 ) + π 2 δ ( 3 4, 1 ). 2 To this end it suffices to prove that for (x 1, x 2 ) not in { (1/4, 1/2), (3/4, 1/2) }, u(x 1, x 2 ) is contained in a set of measure zero and u(x 1, 1/2) is a set of measure π/2 for x 1 {1/4, 3/4}. Indeed it is easy to see that for (x 1, x 2 ) Ω with x 1 1/4 or x 1 3/4 or x 2 1/2, the function u is differentiable and its gradient u satisfies u(x 1, x 2 ) = 1. For those points, the normal mapping of u is always a subset of the unit circle, that is, u(x 1, x 2 ) {p R 2 : p = 1}, and it follows that u(x 1, x 2 ) = 0 for (x 1, x 2 ) Ω with x or x or x So it remains to compute u(x 1, x 2 ) for (x 1, x 2 ) Ω with x 1 = 1/4 or x 1 = 3/4 or x 2 = 1/2. Fix (x 1, x 2 ) Ω such that x 1 = 1/4 or x 1 = 3/4 and let (p 1, p 2 ) u(x 1, x 2 ). By definition of the subdifferential, we have u(y 1, y 2 ) x p 1(y 1 x 1 ) + p 2 (y 2 x 2 ) for 0 < y 1, y 2 < 1. Letting y 1 = x 1 in the above inequality yields y x p 2(y 2 x 2 ) for 0 < y 1 < 1, and it follows that [ 1, 1] if x 2 = 1 2 p 2 x 1 2 (x 2) = 1 if x 2 > if x 2 < 1 2. Consequently for x 1 = 1/4 or x 1 = 3/4 and x 2 1/2, the subdifferential u(x 1, x 2 ) is a subset of the negligible set {p R 2 : p 2 = 1}, and for x 1 = 1/4 or x 1 = 3/4 we have u(x 1, 1/2)

11 11 R [ 1, 1]. Next, we show that { u(x 1, 1 p R 2 2 ) = : p 1 and p 1 0 } if x 1 = 1 4 { p R 2 : p 1 and p 1 0 } if x 1 = 3 4. Indeed for p u(x 1, 1/2), we have (4.1) u(y 1, y 2 ) p 1 (y 1 x 1 ) + p 2 (y ) for all 0 < y 1, y 2 < 1. In particular for y 2 = 1/2, we get p 1 (y 1 x 1 ) 0 for all 1/4 < y 1 < 3/4 and it follows that p 1 0 for x 1 = 1/4 and p 1 0 for x 1 = 3/4. We now consider a line segment through (x 1, 1/2) and direction p and contained in Ω. It has parametric equations (x 1 + tp 1, 1/2 + tp 2 ) for t > 0 sufficiently small. Since u(x 1, 1/2) = 0, by definition of subdifferential we have u(x 1 + tp 1, tp 2) t p 2. On the other hand, when x 1 = 1/4 we have p 1 0 and u(x 1 + tp 1, 1/2 + tp 2 ) t p. Similarly when x 1 = 3/4 we have p 1 0 and u(x 1 + tp 1, 1/2 + tp 2 ) t p. In summary t p 2 u(x 1 + tp 1, tp 2) t p, and it follows that p 1. We have proved that { u(x 1, 1 p R 2 2 ) : p 1 and p 1 0 } if x 1 = 1 4 { p R 2 : p 1 and p 1 0 } if x 1 = 3 4. To conclude the proof, we verify that the reverse inclusion holds as well. We verify it for the case x 1 = 1/4, the argument for x 1 = 3/4 being analogous. Let p R 2 be such that p 1 and p 1 0. Under these assumptions on p, for (y 1, y 2 ) Ω and z = (1/4, 1/2) we have p 1 (y 1 1/4) + p 2 (y 2 1/2) p 2 (y 2 1/2) if y 1 1/4 and p 1 (y ) + p 2(y 2 1 ) = p (y z) p (y z) 2 (y z) = (y )2 + (y )2. It follows that p 1 (y ) + p 2(y 2 1 { y2 2 ) 1 2 u(y 1, y 2 ) if y y y u(y 1, y 2 ) if y 1 < 1 4. Thus the inequality (4.1) holds for all (y 1, y 2 ) Ω and as a result p u(1/4, 1/2). u(1/4, 1/2) and u(3/4, 1/2) have measure π/2. Thus REFERENCES [1] N. E. Aguilera and P. Morin. Approximating optimization problems over convex functions. Numer. Math., 111(1):1 34, [2] G. Awanou. Discrete Aleksandrov solutions of the Monge-Ampère equation. awanou/up.html, [3] G. Awanou. Convergence rate of a stable, monotone and consistent scheme for the Monge-Ampère equation. Symmetry, 8(4):18, [4] G. Awanou. On standard finite difference discretizations of the elliptic Monge-Ampère equation To appear in J Sci Comput doi: /s y. [5] G. Awanou and R. Awi. Convergence of finite difference schemes to the Aleksandrov solution of the Monge-Ampère equation. Acta Applicandae Mathematicae, 144(1):87 98, 2016.

12 12 GERARD AWANOU AND LEOPOLD MATAMBA MESSI [6] J.-D. Benamou and B. D. Froese. A viscosity framework for computing Pogorelov solutions of the Monge-Ampère equation [7] B. Froese and A. Oberman. Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher. SIAM J. Numer. Anal., 49(4): , [8] M. Grant. Disciplined Convex Programming. PhD thesis, Stanford University, [9] M. Grant and S. Boyd. CVX: Matlab software for disciplined convex programming, version com/cvx, Mar [10] C. E. Gutiérrez. The Monge-Ampère equation. Progress in Nonlinear Differential Equations and their Applications, 44. Birkhäuser Boston Inc., Boston, MA, [11] D. Hartenstine. The Dirichlet problem for the Monge-Ampère equation in convex (but not strictly convex) domains. Electron. J. Differential Equations, pages No. 138, 9 pp. (electronic), [12] G. F. LAWLER, Weak convergence of a random walk in a random environment, Comm. Math. Phys., 87 (1982/83), pp [13] P.-L. Lions. Two remarks on Monge-Ampère equations. Ann. Mat. Pura Appl. (4), 142: (1986), [14] K. C. Toh, M. J. Todd, and R. H. Tütüncü. A MATLAB software package for semidefinite programming, version (2): , DEPARTMENT OF MATHEMATICS, STATISTICS, AND COMPUTER SCIENCE, M/C 249. UNIVERSITY OF ILLINOIS AT CHICAGO, CHICAGO, IL , USA address: awanou@uic.edu URL: awanou MATHEMATICAL BIOSCIENCES INSTITUTE, THE OHIO STATE UNIVERSITY, COLUMBUS, OH 43210, USA address: matambamessi.1@mbi.osu.edu URL: