MONGE-AMPÈRE EQUATION ON EXTERIOR DOMAINS

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1 MONGE-AMPÈRE EQUATION ON EXTERIOR DOMAINS JIGUANG BAO, HAIGANG LI, AND LEI ZHANG Abstract. We consider the Monge-Ampère equation det(d 2 u) = f where f is a positive function in R n and f = + O( x β ) for some β > 2 at infinity. If the equation is globally defined on R n we classify the asymptotic behavior of solutions at infinity. If the equation is defined outside a convex bounded set we solve the corresponding exterior Dirichlet problem. Finally we prove for n 3 the existence of global solutions with prescribed asymptotic behavior at infinity. The assumption β > 2 is sharp for all the results in this article.. Introduction It is well known that Monge-Ampère equations are a class of important fully nonlinear equations profoundly related to many fields of analysis and geometry. In the past few decades many significant contributions have been made on various aspects of Monge-Ampère equations. In particular, the Dirichlet problem { det(d 2 u) = f, in D, u = φ on D on a convex, bounded domain D is completely understood through the works of Aleksandrov [], Bakelman [2], Nirenberg [4], Calabi [2], Pogorelov [27, 29, 30], Cheng-Yau [3], Caffarelli-Nirenberg-Spruck [], Caffarelli [7], Krylov [24], Jian-Wang [22], Huang [2], Trudinger-Wang [33], Urbas [35],Savin [3, 32], Philippis-Figalli [26] and the references therein. Corresponding to the traditional Dirichlet problem mentioned above, there is an exterior Dirichlet problem which seeks to solve the Monge-Ampère equation outside a convex set. More specifically, let D be a smooth, bounded and strictly convex subset of R n and let φ C 2 ( D), the exterior Dirichlet problem is to find u to verify det(d 2 u) = f(x), x R n \ D, (.) u C 0 (R n \ D) is a locally convex viscosity solution, u = φ, on D. Date: April 5, Mathematics Subject Classification. 35J96; 35J67. Key words and phrases. Monge-Ampère equation, Dirichlet problem, a priori estimate, maximum principle, viscosity solution. Bao is supported by NSFC (07020) and SRFDPHE ( ), Li is supported by NSFC (07020)(26038)(20029) and SRFDPHE ( ), Zhang is supported in part by NSF Grant

2 2 JIGUANG BAO, HAIGANG LI, AND LEI ZHANG If f and n 3, Caffarelli and Li [9] proved that any solution u of (.) is very close to a parabola near infinity. They solved the exterior Dirichlet problem assuming that u equals φ on D and has a prescribed asymptotic behavior at infinity. For f and n = 2, Ferrer-Martínez- Milán [8, 9] used a method of complex analysis to prove that any solution u of (.) is very close to a parabola plus a logarithmic function at infinity (see also Delanoë [6]). Recently the first two authors [3] solved the exterior Dirichlet problem for f and n = 2. In the first part of this article we solve the exterior Dirichlet problem assuming that f is a perturbation of near infinity: (F A) : f C 0 (R n ), 0 < inf R n f sup R n f <. D 3 f exists outside a compact subset of R n, β > 2 such that lim x x β+ α D α (f(x) ) <, α = 0,, 2, 3. Let M n n be the set of the real valued, n n matrices and A := {A M n n : A is symmetric, positive definite and det(a) = }. Our first main theorem is Theorem.. Let D be a strictly convex, smooth and bounded set, φ C 2 ( D) and f satisfy (FA). If n 3, then for any b R n, A A, there exists c (n, D, φ, b, A, f) such that for any c > c, there exists a unique u to (.) that satisfies (.2) lim sup x min{β,n} 2+k D k (u(x) ( x 2 x Ax + b x + c)) < for k = 0,, 2, 3, 4. If n = 2, then for any b R 2, A A, there exists d R depending only on A, b, φ, f, D such that for all d > d, there exists a unique u to (.) that satisfies (.3) lim sup x x k+σ D k (u(x) ( 2 x Ax + b x + d log x Ax + c d )) < for k = 0,, 2, 3, 4 and σ (0, min{β 2, 2}). c d R is uniquely determined by D, φ, d, f, A, b. The Dirichlet problem on exterior domains is closely related to asymptotic behavior of solutions defined on entire R n. The classical theorem of Jörgens [23], Calabi [2] and Pogorelov [28] states that any convex classical solution of det(d 2 u) = on R n must be a quadratic polynomial. See Cheng-Yau[4], Caffarelli [7] and Jost-Xin [7] for different proofs and extensions. Caffarelli- Li[9] extended this result by considering (.4) det(d 2 u) = f R n where f is a positive continuous function and is not equal to only on a bounded set. They proved that for n 3, the convex viscosity solution u

3 MONGE-AMPÈRE EQUATION 3 is very close to quadratic polynomial at infinity and for n = 2, u is very close to a quadratic polynomial plus a logarithmic term asymptotically. In a subsequent work [0] Caffarelli-Li proved that if f is periodic, then u must be a perturbation of a quadratic function. The second main result of the paper is to extend the Caffarelli-Li results on global solutions in [9]: Theorem.2. Let u C 0 (R n ) be a convex viscosity solution to (.4) where f satisfies (F A). If n 3, then there exist c R, b R n and A A such that (.2) holds. If n = 2 then there exist c R, b R 2, A A such that (.3) holds for d = (f ) and σ (0, min{β 2, 2}). 2π R 2 Corollary.. Let D be a bounded, open and convex subset of R n and let u C 0 (R n \ D) be a locally convex viscosity solution to (.5) det(d 2 u) = f, in R n \ D where f satisfies (FA). Then for n 3, there exist c R, b R n and A A such that (.2) holds. For n = 2, there exist A A, b R n and c, d R such that for k = 0,.., 4 lim sup x x k+σ D k (u(x) ( 2 x Ax + b x + d log x Ax + c)) < holds for σ (0, min{β 2, 2}). As is well known the Monge-Ampère equation det(d 2 u) = f is closely related to the Minkowski problems, the Plateau type problems, mass transfer problems, and affine geometry, etc. In many of these applications f is not a constant. The readers may see the survey paper of Trudinger-Wang [34] for more description and applications. The importance of f not identical to is also mentioned by Calabi in [2]. Next we consider the globally defined equation (.4) and the existence of global solutions with prescribed asymptotic behavior at infinity. Theorem.3. Suppose f satisfies (FA). Then for any A A, b R n and c R, if n 3 there exists a unique convex viscosity solution u to (.4) such that (.2) holds. The following example shows that the decay rate assumption β > 2 in (FA) is sharp in all the theorems. Let f be a radial, smooth, positive function such that f(r) for r [0, ] and f(r) = + r 2 for r > 2. Let r ( s ) u(r) = n n t n n f(t)dt ds, r = x. 0 0 It is easy to check that det(d 2 u) = f in R n. Moreover for n 3, u(x) = 2 x 2 + O(log x ) at infinity. For n = 2, u(x) = 2 x 2 + O((log x ) 2 ) at infinity.

4 4 JIGUANG BAO, HAIGANG LI, AND LEI ZHANG Corresponding to the results in this paper we make the following two conjectures. First we think the analogue of Theorem.3 for n = 2 should also hold. Conjecture : Let n = 2 and f satisfy (FA), then there exists a unique convex viscosity solution u to (.4) such that (.3) holds for d = 2π R (f ) 2 and σ (0, min{β 2, 2}). Conjecture 2: The d in Theorem. is 2π R 2 \D (f ) 2π area(d). These two conjectures are closely related in a way that if conjecture one is proved, then conjecture two follows by the same argument in the proof of Theorem.. The organization of this paper is as follows: First we establish a useful proposition in section two, which will be used in the proof of all theorems. Then in section three we prove Theorem. using Perron s method. Theorem.3 and Theorem.2 are proved in section four and section five, respectively. In the appendix we cite the interior estimates of Caffarelli and Jian-Wang. The proof of all the theorems in this article relies on previous works of Caffarelli [5, 7], Jian-Wang [22] and Caffarelli-Li [9]. For example, Caffarelli-Li [9] made it clear that for exterior Dirichlet problems, convex viscosity solutions are strictly convex. On the other hand for Monge-Ampère equations on convex domains, Pogorelov has a well known example of a notstrictly-convex solution. Besides this, we also use the Alexandrov estimates, the interior estimate of Caffarelli [7] and Jian-Wang [22] in an essential way. 2. A useful proposition Throughout the article we use B r (x) to denote the ball centered at x with radius r and B r to denote the ball of radius r centered at 0. The following proposition will be used in the proof of all theorems. Proposition 2.. Let R 0 > 0 be a positive number, v C 0 (R n \ B R0 ) be a convex viscosity solution of where f v C 3 (R n ) \ B R0 and det(d 2 v) = f v satisfies c 0 f v (x) c 0, R n \ B R0 x R n \ B R0 (2.) D k (f v (x) ) c 0 x β k, x > R 0, k = 0,, 2, 3. Suppose there exists ɛ > 0 such that (2.2) v(x) 2 x 2 c x 2 ɛ, x R 0

5 MONGE-AMPÈRE EQUATION 5 then for n 3, there exist b R n, c R and C(n, R 0, ɛ, β, c 0, c ) such that, (2.3) D k (v(x) 2 x 2 b x c) C/ x min{β,n} 2+k, x > R, k = 0,, 2, 3, 4; where R (n, R 0, ɛ, β, c 0, c ) > R 0 depends only on n, R 0, ɛ, β, c 0 and c. For n = 2, there exist b R 2, d, c R such that for all σ (0, min{β 2, 2}) (2.4) D k (v(x) 2 x 2 b x d log x c) C(ɛ, R 0, β, c 0, c ) x σ+k, x > R, k = 0,, 2, 3, 4 where R > R 0 depends only on ɛ, R 0, β, c 0 and c. Remark 2.. ɛ may be greater than or equal to 2 in Proposition 2.. Proof of Proposition 2.: Proposition 2. is proved in [9] for the case that f outside a compact subset of R n. For this more general case, Theorem 6. in the appendix (A theorem of Caffarelli, Jian-Wang) and Schauder estimates play a central role. First we establish a lemma that holds for all dimensions n 2. Lemma 2.. Under the assumption of Proposition 2., let w(x) = v(x) 2 x 2, then there exist C(n, R 0, ɛ, c 0, c, β) > 0 and R (n, R 0, ɛ, c 0, c, β) > R 0 such that for any α (0, ) D k w(y) C y 2 k ɛ β, k = 0,, 2, 3, 4, y > R (2.5) D 4 w(y ) D 4 w(y 2 ) y y 2 C y α 2 ɛβ α, y > R, y 2 B y (y ) 2 where ɛ β = min{ɛ, β}. Proof of Lemma 2.: For x = R > 2R 0, let and By (2.2) we have v R (y) = ( 4 R )2 v(x + R y), y 2, 4 w R (y) = ( 4 R )2 w(x + R y), y 2. 4 (2.6) v R L (B 2 ) C, w R L (B 2 ) CR ɛ and v R (y) ( 2 y R x y + 8 R 2 x 2 ) = O(R ɛ ), B 2.

6 6 JIGUANG BAO, HAIGANG LI, AND LEI ZHANG Let v R (y) = v R (y) 4 R x y 8 R 2 x 2, clearly D 2 v R = D 2 v R. If R > R with R sufficiently large, the set Ω,v = {y B 2 ; v R (y) } is between B.2 and B n. The equation for v R is (2.7) det(d 2 v R (y)) = f,r (y) := f v (x + R 4 y), on B 2. Immediately from (2.) we have, for any α (0, ) (2.8) f,r L (B 2 ) + Df,R C α (B 2 ) + D 2 f,r C α (B 2 ) CR β. Applying Theorem 6. on Ω,v Using (2.7) and (2.8) we have D 2 v R C α (B. ) = D 2 v R C α (B. ) C. I (2.9) C D2 v R CI on B. for some C independent of R. Rewrite (2.7) as a R ij ij v R = f,r, B 2 where a R ij = cof ij(d 2 v R ). Clearly by(2.9) a R ij is uniformly elliptic and By Schauder estimates a R ij C α ( B. ) C. (2.0) v R C 2,α ( B ) C( v R L ( B. ) + f,r C α ( B 2 ) ) C. For any e S n, apply e to both sides of (2.7) (2.) a R ij ij ( e v R ) = e f,r. Since a R ij, ev R and e f,r are bounded in C α norm, we have (2.2) v R C 3,α ( B ) C, which implies (2.3) a R ij C,α ( B ) C. The difference between (2.7) (with v R replaced by v R ) and det(i) = gives (2.4) ã ij ij w R = f,r (y) = O(R β ) where ã ij (y) = 0 cof ij(i + td 2 w R (y))dt. By (2.9) and (2.2) I C ã ij CI, on B., ã ij C,α ( B ) C. Thus Schauder s estimate gives (2.5) w R C 2,α (B ) C( w R L ( B. ) + f,r C α ( B ) ) CR ɛ β.

7 Going back to (2.) and rewriting it as We obtain, by Schauder s estimate, MONGE-AMPÈRE EQUATION 7 a R ij ij ( e w R ) = e f,r. (2.6) w R C 3,α ( B /2 ) C( w R L ( B 3/4 ) + Df,R C α ( B 3/4 ) ) CR ɛ β. Since ije w R = ije v R, we also have D 3 v R C α ( B /2 ) CR ɛ β. By differentiating on (2.) with respect to any e S n we have (2.6) gives a R ij ij ( ee w R ) = ee f,r e a R ij ij e w R. e a R ij ij e w R C α ( B /2 ) CR 2ɛ β. Using (2.8), ee f,r C α ( B ) CR β and Schauder s estimate we have (2.7) w R C 4,α ( B /4 ) CR ɛ β, which implies (2.5). Lemma 2. is established. Next we prove a lemma that improves the estimates in Lemma 2.. Lemma 2.2. Under the same assumptions of Lemma 2. and let R be the large constant determined in the proof of Lemma 2.2. If in addition 2ɛ <, then for n 3 D j w(x) C x 2 2ɛ j, x > 2R, j = 0,, 2, 3, 4 D 4 w(y ) D 4 w(y 2 ) y y 2 α C y 2 2ɛ α, y > 2R, y 2 B y /2(y ) where α (0, ). For n = 2 and any ɛ < 2ɛ < D j w(x) C x 2 ɛ j, x > 2R, j = 0,, 2, 3, 4 D 4 w(y ) D 4 w(y 2 ) y y 2 α C y 2 ɛ α, y > 2R, y 2 B y /2(y ). Proof of Lemma 2.2: Apply k to det(d 2 v) = f v we have (2.8) a ij ij ( k v) = k f v where a ij = cof ij (D 2 v). Lemma 2. implies and for any α (0, ) a ij (x) δ ij C x ɛ, Da ij(x) C x +ɛ, x > R Da ij (x ) Da ij (x 2 ) x x 2 α C x ɛ α, x > 2R, x 2 B x /2(x ).

8 8 JIGUANG BAO, HAIGANG LI, AND LEI ZHANG Then apply l to (2.8) and let h = kl v a ij ij h = kl f v l a ij ijk v. We further write the equation above as (2.9) h = f 2 := kl f v l a ij ijk v (a ij δ ij ) ij h. By (2.) and Lemma 2., for any α (0, ) f 2 (x) C x 2 2ɛ x 2R, (2.20) f 2 (x ) f 2 (x 2 ) C x x 2 α x, x 2+2ɛ+α 2 B x /2(x ), x 2R. Note that by Lemma 2. h (x) δ kl as x. If n 3, let h 2 (x) = x y 2 n f 2 (y)dy R n \B R n(n 2)ω n where ω n is the volume of the unit ball in R n. If n = 2, let h 2 (x) = (log x y log x )f 2 (y)dy. 2π R 2 \B R In either case h 2 = f 2. By elementary estimate it is easy to get { (2.2) D j C x h 2 (x) 2ɛ j, x > 2R, j = 0,, n 3, C x ɛ j, x > 2R, j = 0,, n = 2 where ɛ is any positive number less than 2ɛ. Indeed, for each x, let E := {y R n \ B 2R, y x /2, }, E 2 := {y R n \ B 2R, y x x /2, }, E 3 = (R n \ B 2R ) \ (E E 2 ). Then it is easy to get (2.2). For the estimate of D 2 h 2 we claim that given α (0, ), if n 3 D j h 2 (x) C x 2ɛ j, j = 0,, 2, x > 2R, (2.22) D 2 h 2 (x ) D 2 h 2 (x 2 ) C x x 2 α x, x 2+2ɛ+α 2 B x (x ), x > 2R. Replacing 2ɛ by ɛ we get the corresponding estimates of D 2 h 2 for n = 2. The way to obtain (2.22) is standard. Indeed, for each x 0 R n \ B 2R, let R = x 0, we set h 2,R (y) = h 2 (x 0 + R 4 y), f 2,R(y) = R2 6 f 2(x 0 + R y), y 2. 4 By (2.20) f 2,R C α (B ) CR 2ɛ. Therefore Schauder estimate gives h 2,R C 2,α (B ) C( h 2,R L (B 2 ) + f 2,R C α (B 2 )) CR 2ɛ, which is equivalent to (2.22). The way to get the corresponding estimate for n = 2 is the same. Now we have (h h 2 ) = 0, R n \ B 2R. 2

9 MONGE-AMPÈRE EQUATION 9 Since we know h δ kl h 2 0 at infinity. For n 3, by comparing with a multiple of x 2 n we have h (x) δ kl h 2 (x) C x 2 n, x > 2R. By the estimate on h 2 we have Correspondingly h (x) δ kl C x 2ɛ, x > 2R. D j w(x) C x 2 j 2ɛ, x > 2R, j = 0,, 2, n 3. For n = 2 we have (2.23) h (x) δ kl h 2 (x) C x, x > 2R. Indeed, let h 3 (y) = h ( y y 2 ) δ kl h 2 ( y y 2 ), then h 3 = 0 in B /2R \ {0} and lim y 0 h 3 (y) = 0. Therefore h 3 (y) C y near 0. (2.23) follows. By fundamental theorem of calculus, D j w(x) C x 2 j ɛ, x > 2R, j = 0,, 2, n = 2. Finally we apply Lemma 2. to obtain the estimates on the third and fourth derivatives. Lemma 2.2 is established. Case one: n 3. Let k 0 be a positive integer such that 2 k 0 ɛ < and 2 k0+ ɛ > ( we choose ɛ smaller if necessary to make both inequalities hold). Let ɛ = 2 k 0 ɛ, clearly we have < 2ɛ < 2. Applying Lemma 2.2 k 0 times we have (2.24) D k w(x) C x 2 ɛ k, k = 0,.., 4, x > 2R D 4 w(x ) D 4 w(x 2 ) x x 2 α C x 2 ɛ α, x > 2R, x 2 B x /2(x ). Let h and f 2 be the same as in Lemma 2.2. Then we have f 2 (x) C x 2 2ɛ x 2R, f 2 (x ) f 2 (x 2 ) x x 2 α C x 2+2ɛ +α, x 2 B x /2(x ), x 2R. Constructing h 2 as in Lemma 2.2 ( the one for n 3) we have D j h 2 (x) C x 2ɛ j, j = 0,, 2, x > 2R, (2.25) D 2 h 2 (x ) D 2 h 2 (x 2 ) C x x 2, x α x 2+2ɛ +α 2 B x (x ), x > 2R. As in the proof of Lemma 2.2 by (2.25) we have Since 2ɛ > h (x) h 2 (x) C x 2 n, x > 2R. h (x) h 2 (x) + C x 2 n C x. 2

10 0 JIGUANG BAO, HAIGANG LI, AND LEI ZHANG By Theorem 4 of [20], m w(x) c m for some c m R as x. Let b R n be the limit of w and w (x) = w(x) b x. The equation for w can be written as ( for e S n ) (2.26) a ij ij ( e w ) = e f v. By (2.24) the equation above can be written as (2.27) ( e w ) = f 3 := e f v (a ij δ ij ) ije w, x > 2R. and we have f 3 (x) C( x β + x 2ɛ ) C x 2ɛ, x > 2R f 3 (x ) f 3 (x 2 ) x x 2 α C x 2ɛ α, x > 2R, x 2 B x /2(x ). Let h 4 solve h 4 = f 3 and the construction of h 4 is similar to that of h 2. Then we have D j h 4 (x) C x 2ɛ j, x > 2R, j = 0,, 2, D 2 h 4 (x ) D 2 h 4 (x 2 ) x x 2 α C x 2ɛ α, x > 2R, x 2 B x /2(x ). Since e w h 4 0 at infinity, we have (2.28) e w (x) h 4 (x) C x 2 n, x > R. Therefore we have obtained w (x) C x 2ɛ on x > R. Using fundamental theorem of calculus Lemma 2. applied to w gives w (x) C x 2 2ɛ, j = 0,, x > R. D j w (x) C x 2 j 2ɛ, j = Going back to (2.27), now the estimate for f 3 becomes f 3 (x) C x β + C x 4ɛ, x > 2R, f 3 (x ) f 3 (x 2 ) x x 2 α C( x β α + x 4ɛ α ), x > 2R, x 2 B x /2(x ). The new estimate of h 4 is h 4 (x) C( x β + x 4ɛ ), x > 2R. As before (2.28) holds. Consequently w (x) C( x 2 n + x 4ɛ ) C x, x > 2R. By Theorem 4 of [20], w c at infinity. Let w 2 (x) = w(x) b x c. Then we have w 2 (x) C for x > 2R. Lemma 2. applied to w 2 gives (2.29) D k w 2 (x) C x k, k = 0,, 2, 3, x > 2R.

11 The equation for w 2 can be written as MONGE-AMPÈRE EQUATION det(i + D 2 w 2 (x)) = f v. Taking the difference between this equation and det(i) = we have where ã ij satisfies ã ij ij w 2 = f v, x > 2R D j (ã ij (x) δ ij ) C x 2 j, x > 2R, j = 0,. Using (2.29) this equation can be written as w 2 = f 4 := f v (ã ij δ ij ) ij w 2, x > 2R. (2.29) further gives f 4 (x) C( x β + x 4 ), x > 2R, f 4 (x ) f 4 (x 2 ) x x 2 α C( x β α + x 4 α ), x > 2R, x 2 B x /2(x ). Let h 5 be defined similar to h 2. Then h 5 solves h 5 = f 4 in R n \ B 2R and satisfies h 5 (x) C( x 2 β + x 2 ). As before we have which gives w 2 (x) h 5 (x) C x 2 n, x > 2R, (2.30) w 2 (x) C( x 2 n + x 2 β + x 2 ), x > 2R. If x 2 > x 2 n + x 2 β we can apply the same argument as above finite times to remove the x 2 from (2.30). Eventually by Lemma 2. we have (2.3). Proposition 2. is established for n 3. Case two: n = 2 As in the case for n 3 we let k 0 be a positive integer such that 2 k 0 ɛ < and 2 k 0+ ɛ > ( we choose ɛ smaller if necessary to make both inequalities hold). Let ɛ < 2 k 0 ɛ and we let < 2ɛ < 2. Applying Lemma 2.2 k 0 times then (2.24) holds. Consider the equation for w. By taking the difference between the equation for v and det(i) = we have ã ij ij w = f v. We further write the equation above as By (2.24) Let w = f 5 := f v (ã ij δ ij ) ij w. f 5 (x) C x 2ɛ, x > R. h 6 (x) = 2π R 2 \B R (log x y log x )f 5 (y)dy.

12 2 JIGUANG BAO, HAIGANG LI, AND LEI ZHANG Then elementary estimate gives h 6 (x) C x ɛ 2, x > R for some ɛ 2 (0, ). Since w h 6 is harmonic on R 2 \ B R and w h 6 = O( x 2 ɛ ), there exist b R 2 and d, d 2 R such that (2.3) w(x) h 6 (x) = b x + d log x + d 2 + O(/ x ) x > 2R. Equation (2.3) is standard. For the convenience of the readers we include the proof. Let z l (r) be the projection of w h 6 on sin lθ for l =, 2,... Then z l satisfies z l (r) + r z l2 l (r) r 2 z l(r) = 0, r > 2R. Clearly z l (r) = c l r l + c 2l r l. Since z l (r) Cr 2 ɛ we have c l = 0 for all l 2. Thus z l (r) = c 2l r l. Let C be a constant such that max B2R w h 6 C. Then z l (2R ) C, which gives c 2l C(2R ) l. The estimate for the projection of w h 6 over cos lθ for l 2 is the same. The term d log x +d 2 comes from the projection onto. The projection onto cos θ and sin θ gives b x. (2.3) is established. Let w (x) = w(x) b x. Then w (x) C x ɛ 2. Apply Lemma 2. D k w (x) C x ɛ 2 k, k = 0,.., 4, x > 2R. The equation for w can be written as Let Then w = O( x β ) + O( x 2ɛ 2 4 ). h 7 (x) = (log x y log x ) w (y)dy. 2π R 2 \B 2R h 7 (x) C( x 2 β+ɛ + x 2ɛ 2 2+ɛ ) for ɛ > 0 arbitrarily small. Since w h 7 is harmonic on R 2 \ B 2R and w (x) h 7 (x) = O( x ɛ 2 ), we have, for some d, c R Using the estimates on h 7 we have w (x) h 7 (x) = d log x + c + O(/ x ). (2.32) w (x) = d log x + c + O( x 2ɛ 2 2+ɛ ) + O( x 2 β+ɛ ). To obtain (2.4) we finally let and v (x) = v(x) b x c H(x) = 2 x 2 + d log x.

13 MONGE-AMPÈRE EQUATION 3 Clearly det(d 2 v (x)) = f v (x) and det(d 2 H(x)) = d2 x 4. Let w 2 (x) = v (x) H(x). By (2.32) we already have w 2 (x) C x ɛ 3, x > 2R for some ɛ 3 > 0. Using Theorem 6. as well as Schauder estimate as in the proof of Lemma 2. we obtain (2.33) D k w 2 (x) C x ɛ 3 k Thus the equation of w 2 can be written as Let Then x > 2R, k = 0,, 2. w 2 (x) = O( x 4 2ɛ 3 ) + O( x β ). h 8 (x) = (log x y log x ) w 2 (y)dy. 2π R 2 \B R (2.34) D j h 8 (x) = O( x 2 j + x ɛ 5+2 β j ), j = 0,, x > R for all ɛ 5 > 0. Then we have w 2 (x) h 8 (x) = O( x 2 ) because of (2.33), (2.34) and the argument in the proof of (2.3). Consequently w 2 (x) = O( x 2 + x ɛ 6+2 β ), x > R for all ɛ 6 > 0. The estimates on the derivatives of w 2 can be obtained by Lemma 2.. Proposition 2. is established for n = 2 as well. 3. Proof of Theorem. Without loss of generality we assume that B 2 D B r. First we prove a lemma that will be used in the proof for n 3 and n = 2. Lemma 3.. There exists c (n, φ, D) such that for every ξ D, there exists w ξ such that det(d 2 w ξ (x)) f(x) R n \ D, w ξ (ξ) = φ(ξ), w ξ (x) < φ(x), x D, x ξ, w ξ (x) 2 x 2 + c, x (R n \ D) B(0, 0diam(D)). Proof of Lemma 3.: Let f be a smooth radial function on R n such that f > f on R n \ D and f satisfies (FA). Let z(x) = x 0 ( s 0 nt n f (t)dt) n ds. Then det(d 2 z(x)) = f (x) on R n and z(x) { C, n 3, 2 x 2 C log(2 + x ), n = 2 x R n. Since D is strictly convex, we can put ξ as the origin using a translation and a rotation and then assume that D stays in {x n > 0}. Assume that the

14 4 JIGUANG BAO, HAIGANG LI, AND LEI ZHANG boundary around ξ is described by x n = ρ(x ) where x = (x,.., x n ). By the strict convexity we assume ρ(x ) = B αβ x α x β + o( x 2 ) 2 α,β n where (B αβ ) δi for some δ > 0. By subtracting a linear function from z we obtain z ξ that satisfies { det(d 2 z ξ ) f, R n \ D, z ξ (0) = φ(ξ), z ξ (0) = φ(ξ) and Next we further adjust z ξ by defining z ξ (x) 2 x 2 C x, x R n \ D. w ξ (x) = z ξ (x) A ξ x n for A ξ large to be determined. When evaluated on D near 0, w ξ (x, ρ(x )) φ(x, ρ(x )) C x 2 A ξ ρ(x ). Therefore for x δ for some δ small we have w ξ (x, ρ(x )) < φ(x, ρ(x )). For x > δ, the convexity of D yields x n δ 3, x D \ {(x, ρ(x )) : x < δ.}. Then by choosing A ξ possibly larger (but still under control) we have w ξ (x) < φ(x) for all x D. Clearly A ξ has a uniform bound for all ξ Ω. Lemma 3. is established. Let w(x) = max { w ξ (x) ξ D }. It is clear by Lemma 3. that w is a locally Lipschitz function in B 2 r \ D, and w = ϕ on D. Since w ξ is a smooth convex solution of (.), w is a viscosity subsolution of (.) in B 2 r \ D. Let c be the constant determined in Lemma 3.. Then we have w(x) 2 x 2 + c, B 2 r \ D. We finish the proof of Theorem. in two cases. Case one: n 3. Clearly we only need to prove the existence of solutions for A = I and b = 0, as the general case can be reduced to this case by a linear transformation. Let f and f be smooth, radial functions such that f < f < f in R n \ D and suppose f and f satisfy (FA). For d > 0 and β, β 2 R, set u d (x) = β + r r ( s nt n f(t)dt + d ) n ds, r = x > 2,

15 and Clearly and MONGE-AMPÈRE EQUATION 5 r ( s ) u d (x) = β 2 + nt n n f(t)dt + d ds, r = x > 2. 2 On the other hand, det(d 2 u d ) = f f, det(d 2 u d ) = f f, R n \ D, R n \ D. (3.) u d (x) β, in B r \ D, d > 0. and (3.2) u d (x) β 2, in B r \ D, d > 0. Let β := min { w(x) x B r \ D } < min ϕ, D β 2 := max ϕ +. D This shows that u d and u d are continuous convex subsolution and supersolution of (.5), respectively. By the definition of u d by choosing d large enough, say d d 0, we can make u d > w(x) +, x = r +. By (3.) and the above, the function u d, x r +, u,d (x) = w(x), x B r \ D, max{w(x), u d }, x B r+ \ B r is a viscosity subsolution of (.5) if d d 0. Next we consider the asymptotic behavior of u d and ū d when d is fixed. Using (FA) it is easy to obtain u d (x) = 2 x 2 + µ (d) + O( x 2 min{β,n} ), and where and u d (x) = 2 x 2 + µ 2 (d) + O( x 2 min{β,n} ), µ (d) = β r2 2 + µ 2 (d) = β r 2 ( ( s ( ( s nt n f(t)dt + d ) n s ) ds, ) ) nt n n f(t)dt + d s ds.

16 6 JIGUANG BAO, HAIGANG LI, AND LEI ZHANG It is easy to see that µ (d) and µ 2 (d) are strictly increasing functions of d and (3.3) lim d µ (d) =, and lim d µ 2(d) =. Let c = µ (d 0 ), recall that for d > d 0, u,d is a viscosity subsolution. For every c > c, there exists a unique d(c) such that (3.4) µ (d(c)) = c. So u d(c) satisfies (3.5) u d(c) (x) = 2 x 2 + c + O ( x 2 min{β,n}), as x. Also there exists d 2 (c) such that µ 2 (d 2 (c)) = c and (3.6) u d2 (c)(x) = ( 2 x 2 + c + O x 2 min{β,n}), as x. By (3.5) and (3.6) ( ) lim u d(c) (x) u d2 (c)(x) = 0. x On the other hand, by the definition of β we have ū d2 (c) > u,d(c) on D. Thus, in view of the comparison principle for smooth convex solutions of Monge-Ampère, (see []), we have (3.7) u,d(c) ū d2 (c), on R n \ D. For any c > c, let S c denote the set of v C 0 (R n \D) which are viscosity subsolutions of (.5) in R n \ D satisfying (3.8) v = ϕ, on D, and (3.9) u,d(c) v ū d2 (c), in R n \ D. We know that u,d(c) S c. Let u(x) := sup {v(x) v S c }, x R n \ D. Then u is convex and of class C 0 (R n \ D). By (3.5), and the definitions of u,d(c) and ū d2 (c) (3.0) u(x) u,d(c) (x) = ( 2 x 2 + c + O x 2 min{β,n}), as x and u(x) u d2 (c)(x) = ( 2 x 2 + c + O x 2 min{β,n}). The estimate (.2) for k = 0 follows. Next, we prove that u satisfies the boundary condition. It is obvious from the definition of u,d(c) that lim inf x ξ u(x) lim x ξ u,d(c) (x) = ϕ(ξ), ξ D.

17 So we only need to prove that Let ω + c MONGE-AMPÈRE EQUATION 7 lim sup u(x) ϕ(ξ), x ξ ξ D. C 2 (B r \ D) be defined by ω c + = 0, in B r+ \ D, ω c + = ϕ, on D, ω c + = max u d2 (c), on B r+. B r+ It is easy to see that a viscosity subsolution v of (.5) satisfies v 0 in viscosity sense. Therefore, for every v S c, by v ω c + on (B r \ D), we have v ω c + in B r \ D. It follows that and then u ω + c in B r \ D, lim sup u(x) lim ω c + (x) = ϕ(ξ), x ξ x ξ ξ D. Finally, we prove u is a solution of (.). For x R n \ D, fix some ɛ > 0 such that B ɛ ( x) R n \ D. By the definition of u, u ū. We claim that there is a convex viscosity solution to ũ C 0 (B ɛ ( x)) to { det(d 2 ũ) = f, x B ɛ ( x), ũ = u, x B ɛ ( x). Indeed, let φ k be a sequence of smooth functions on B ɛ ( x) satisfying u φ k u + k. Let f k be a sequence of smooth positive functions tending to f and f k f. Let ψ k be the convex solution to { det(d 2 ψ k ) = f k B ɛ ( x), ψ k = φ i on B ɛ ( x). Clearly ψ k u. On the other hand, let h k be the harmonic function on B ɛ ( x) with h k = φ k on B ɛ ( x). Then we have u k h k. Therefore ψ k is uniformly bounded over any compact subset of B ɛ ( x). ψ k is also uniformly bounded over all compact subsets of B ɛ ( x) by the convexity. Thus ψ k converges along a subsequence to ũ in B ɛ ( x). By the closeness between h k to u on B ɛ ( x), ũ can be extended as a continuous function to B ɛ ( x). By the maximum principle, u ũ ū d2 (c) on B ɛ. Define w(y) = { ũ(y), if y B ɛ, u(y), if y R 2 \ (D B ɛ ( x)).

18 8 JIGUANG BAO, HAIGANG LI, AND LEI ZHANG Clearly, w S c. So, by the definition of u, u w on B ɛ ( x). It follows that u ũ on B ɛ ( x). Therefore u is a viscosity solution of (.). We have proved (.2) for k = 0. The estimates of derivatives follow from Proposition 2.. Theorem. is established for n 3. Case two: n = 2. As in case one we let f be a radial function such that f( x ) f(x) in R 2 \ D, and f also satisfies (FA). Let r ( s ) u d (x) = β + 2t f(t)dt 2 + d ds for d 0 and r >. Here we choose β = min D φ. Clearly r u d (x) < w(x) B r \ D, d 0. Then we choose d large so that for all d d, u d (x) > w(x) on B r+. Let w(x), B r \ D u,d (x) = max{w(x), u d }, B r+ \ B r, u d, R 2 \ B r+. Then u,d is a convex viscosity subsolution of (.5). Let A d = d + Then elementary computation gives 2t( f(t) )dt. u d (x) = 2 x 2 + A d log x + O(). Next we let f be a radial function such that f( x ) f(x) for x R 2 \ D. Suppose f also satisfies (FA) and is positive and smooth on R 2. Let r ( s ) 2 ū d (x) = β 2 + 2tf(t)dt + d ds. Let 2 L d = d + 2t(f(t) )dt. Then the asymptotic behavior of ū d at infinity is ū d (x) = 2 x 2 + L d log x + O(). Thus for all d > d, we can choose d such that L d = A d. Then we choose β such that ū d > φ on D and ū d > u d at infinity. As in case one, by taking the supremum of subsolutions we obtain a solution u that is equal to φ on D and By Proposition 2. u(x) = 2 x 2 + A d log x + O(). u(x) = 2 x 2 + A d log x + c + o().

19 MONGE-AMPÈRE EQUATION 9 The following lemma says the constant term is uniquely determined by other parameters. Lemma 3.2. Let u, u 2 be two locally convex smooth functions on R 2 \ D where D satisfies the same assumption as in Theorem.. Suppose u and u 2 both satisfy { det(d 2 u) = f in R 2 \ D, u = φ, on D with f satisfying (FA) and for the same constant d (3.) u i (x) 2 x 2 d log x = O(), x R 2 \ D, i =, 2. Then u u 2. Proof of Lemma 3.2: By Proposition 2. we see that when (3.) holds, we have D 2 u i (x) = I + O( x 2+ɛ ), i =, 2 for ɛ > 0 small and x large. For the proof of this lemma we only need (3.2) D 2 u i (x) = I + O( x 3 2 ), x >, i =, 2. By Proposition 2., u i (x) = 2 x 2 + d log x + c i + O(/ x σ ), i =, 2 for σ (0, min{β 2, 2}). Without loss of generality we assume c > c 2. If c = c 2 we know u u 2 by maximum principle. Since u = u 2 on D, we have, u > u 2 in R 2 \ D. Let w = u u 2, then w satisfies where a ij (x) = a ij ij w = 0, 0 R 2 \ D cof ij (td 2 u + ( t)d 2 u 2 )dt. By the assumption of Lemma 3.2 and (3.2), a ij is uniformly elliptic and (3.3) a ij (x) = δ ij + O( x 3 2 ), x R 2 \ D. Let a 0 < 2 a be positive constants to be determined. We set h ɛ = ɛ log( x a 0 ) over a < x <. Direct computation shows, by (3.3) that a ij ij h ɛ = h ɛ + (a ij δ ij ) ij h ɛ ɛa 0 ( x a 0 ) 2 x + Cɛ x 7/2, 4ɛa 0 x 3 + Cɛ x 7/2, x > a > a 0. By choosing a 0 sufficiently large and a > 2a 0 we have a ij ij h ɛ < 0, a < x <.

20 20 JIGUANG BAO, HAIGANG LI, AND LEI ZHANG Let R > a and M R = max x =R w. Let v = w M R, then clearly for all ɛ > 0, h ɛ is greater than v on B R and at infinity. Thus for any compact subset K R 2 \ B R, v < h ɛ. Let ɛ 0 we have w(x) M R, x R. Taking any R > R, we have max x =R w M R. Strong maximum principle implies that either max x =R w < M R for all R > R or w is a constant. w is not a constant, therefore we have max x =R w < M R for all R > R. However, this means over the region B R \ D, the maximum of w is attained at an interior point, a contradiction to the elliptic equation that w satisfies. Thus Lemma 3.2 is established. Lemma 3.2 uniquely determines the constant in the expansion, then by Proposition 2. we obtain (.3). Thus Theorem. for the case n = 2 is established. 4. Proof of Theorem.3 We only need to consider the existence part as the uniqueness part follows immediately from maximum principles. For the existence part we only need to consider the case that A = I, b = 0 and c = 0, because the general case can be reduced to this case by a linear transformation. Consider u R that solves det(d 2 u R ) = f, B R, (4.) u R = R2 2, B R. We shall bound u R above and below by two radial functions. Let h be a smooth radial function, then at the point ( x, 0,..., 0) D 2 h(x) = diag(h (r), h (r)/r,..., h (r)/r), r = x. Thus det(d 2 h)(x) = h (r)(h (r)/r) n. We first construct a subsolution h (r): Let f be a radial function such that f > f and f satisfies (FA). h (r) = r 0 ( s 0 nt n f(t)dt) n ds. Clearly det(d 2 h ) = f in R n and since f(t) = + O(t β ) it is easy to verify that h (r) = 2 x 2 + O(). Next we construct a super solution. Let f be a radial function less than f(x) and f also satisfy (FA), h + (r) = r 0 ( s 0 nt n f(t)dt) n ds.

21 MONGE-AMPÈRE EQUATION 2 Similarly we have det(d 2 h + ) = f in R n and h + (r) = 2 r2 + O() for r large. Let β be a constant such that h ( x ) + β 2 x 2, β + be a constant such that h + ( x ) + β + 2 x 2. Then by maximum principle (4.2) h (r) + β u R (x) h + (r) + β +, x R. Let R and the sequence u R converges to a global solution u that satisfies det(d 2 u) = f in R n and u 2 x 2 = O(). For this convergence, we use the fact that for any K R n, u R (x) 2 x 2 C(K) and by Caffarelli s C,α estimate [6], u R L (K) C(K). Thus u R converges to a convex viscosity solution u to det(d 2 u) = f in R n with the property that u(x) 2 x 2 C, R n. By Proposition 2., there exists a c R such that lim x x min{β,n} 2+k ( D k (u(x) 2 x 2 c ) ) < for k = 0,, 2, 3, 4. After a translation the solution with the desired asymptotic behavior can be found. Theorem.3 is established. 5. The proof of Theorem.2 Without loss of generality we assume u(0) = 0 = min R n u. The goal is to show that there exists a linear transformation T such that v = u T satisfies (2.2). Then we employ Proposition 2. to finish the proof. The proof of v satisfying (2.2) is by the argument of Caffarelli-Li. Suppose c 0 inf R n f sup R n f < c 0, only under this assumption it is proved in [9] that for M large and there exists a M A such that Ω M := {x R n ; u(x) < M } (5.) B R/C a M (Ω M ) B CR, where R = M and C > is a constant independent of M. Let Then B /C O B C. Set O := {y; a M (Ry) Ω M}. then we have (5.2) Let ξ solve ξ(y) := R 2 u(a M (Ry)), { det(d 2 ξ) = f(a M (Ry)), in O, ξ =, on O. { det(d 2 ξ) =, in O, ξ =, on O.

22 22 JIGUANG BAO, HAIGANG LI, AND LEI ZHANG By Pogorelov s estimate C I D2 ξ CI, D 3 ξ(x) C, x O, dist(x, O) δ. We claim that there exists C > 0 independent of M such that (5.3) ξ(x) ξ(x) C/R, x O. Indeed, by the Alexandrov estimate ([8]) ( min(ξ ξ) C det(d 2 (ξ ξ)) Ō S + where On S + so the concavity of det n Thus S + := {x O; D 2 (ξ ξ) > 0 }. D 2 ξ 2 = D2 (ξ ξ) 2 + D2 ξ 2, ) /n on positive definite symmetric matrices implies det(d 2 (ξ ξ)) n f(a M (Ry)) n. ( min(ξ ξ) C f(a M (Ry)) n n dy) n. Ō S + Let z = a M (Ry), i.e. a Mz = Ry then dz = R n dy ( ) ( f(a M (Ry)) n ) n n dy S + R B CR f(z) n n dz ) n. By the assumption (FA) the integral is finite, thus we have proved that min(ξ ξ) C/R, x O Ō Similarly we also have min Ō ( ξ ξ) C/R. (5.3) is proved. Next we set E M := {x; (x x) D 2 ξ( x)(x x) } where x is the minimum of ξ. By Theorem of [5] x is the unique minimum point of ξ. Then by the same argument as in [9] we have the following: There exist k and C depending only on n and f such that for ɛ = 0, M = 2(+ɛ)k, 2 k M 2 k, ( 2M R 2 3ɛk C2 2 ) 2 EM R a M(Ω M ) ( 2M R 2 3ɛk + C2 2 ) 2 EM, k k, which is 2M ( C 2 ɛk/2 )E M a M (Ω M ) 2M ( + C 2 ɛk/2 )E M.

23 MONGE-AMPÈRE EQUATION 23 Let Q be a positive definite matrix such that Q 2 = D 2 ξ( x), O be an orthogonal matrix such that T k := OQ k a M is upper triangular. Then clearly det(t k ) = and by Proposition 3.4 of [9] we have Let v = u T, then clearly T k T C2 ɛk 2. det(d 2 v(x)) = f(t x). For v and some k large we have 2M ( C 2 ɛk/2 )B {x; v(x) < M } Consequently 2M ( + C 2 ɛk/2 )B M 2 k. (5.4) v(x) 2 x 2 C x 2 ɛ. Clearly f(t ) also satisfies (FA). Proposition 2. gives the asymptotic behavior of u and the estimates on its derivatives. the constant d in the estimate in two dimensional spaces is determined similarly as in [9]. Theorem.2 is established. Remark 5.. Corollary. follows from Theorem.2 just like in [9] so we omit the proof. 6. Appendix: Interior estimate of Caffarelli and Jian-Wang The following theorem is a combination of the interior estimate of Caffarelli [7] and an improvement by Jian-Wang [22]. Theorem 6.. (Caffarelli, Jian-Wang) Let u C 0 (Ω) be a convex viscosity solution of det(d 2 u) = f, Ω, u = 0 on Ω, where Ω is a convex bounded domain satisfying B Ω B n. Assume that f is Dini continuous on Ω and c 0 f c 0, Ω. Then u C 2 (B /2 ) and x, y B /2 (6.) D 2 u(x) D 2 u(y) C ( d + d 0 ω f (r) r ω f (r)) + d d r 2 where d = x y, C > 0 depends only on n and c 0, ω f is the oscillation function of f defined by ω f (r) := sup{ f(x) f(y) : x y r}.

24 24 JIGUANG BAO, HAIGANG LI, AND LEI ZHANG It follows that (i) If f is Dini continuous, then u C 2 (B /2 ), and the modulus of convexity of D 2 u can be estimated by (6.). (ii) If f C α (Ω) and α (0, ), then (iii) If f C 0, (Ω), then D 2 u C α (B /2 ) C ( + f C α (Ω) ). α( α) D 2 u(x) D 2 u(y) Cd ( + f C 0, (Ω) log d ). Here we recall that f is Dini continuous if the oscillation function ω f satisfies 0 ω f (r)/rdr <. Remark 6.. Note that in Caffarelli s interior estimate u = 0 is assumed on Ω. Since Ω is very close to a ball, by [5, 6] u is strictly convex in Ω. But there is no explicit formula that describes how the higher order derivatives of u depend on lf. In Jian-Wang s theorem, this dependence is given as in (6.) but instead of assume u = 0 on Ω, they assumed u is strictly convex and their constant depends on the strict convexity. We feel the way that Theorem 6. is stated is more convenience for application. We only used the (ii) and (iii) of Theorem 6. in this article. References [] A. D. Aleksandrov, Dirichlet s problem for the euqation Det z ij = φ I, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 3 (958), [2] I. J. Bakelman, Generalized solutions of Monge-Ampère equations (Russion). Dokl. Akad. Nauk SSSR (N. S. ) 4 (957), [3] J. Bao, H. Li, On the exterior Dirichlet problem for the Monge-Ampère equation in dimension two, Nonlinear Anal. 75 (202), no. 8, [4] L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large. Comm. Pure Appl. Math. 6 (953), [5] L. A. Caffarelli, A localization property of viscosity solutions to the Monge Ampère equation and their strict convexity, Ann. of Math. 3 (990) [6] L. A. Caffarelli, Some regularity properties of solutions of Monge Ampère equation. Comm. Pure Appl. Math. 44 (99), no. 8-9, [7] L. A. Caffarelli, Interior W 2,p estimates for solutions of the Monge-Ampère equation, Ann. of Math. 3 (990) [8] L. A. Caffarelli, X. Cabre, Fully nonlinear elliptic equations. American Mathematical Society Colloquium Publications, 43. American Mathematical Society, Providence, RI, 995. vi+04 pp. [9] L. A. Caffarelli, Y. Y. Li, An extension to a theorem of Jörgens, Calabi, and Pogorelov. Comm. Pure Appl. Math. 56 (2003), no. 5, [0] L. A. Caffarelli, Y. Y. Li, A Liouville theorem for solutions of the Monge-Ampère equation with periodic data. Ann. Inst. H. Poincar Anal. Non Linaire 2 (2004), no., [] L. A. Caffarelli, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear secondorder elliptic equations. I. Monge-Ampère equation. Comm. Pure Appl. Math. 37 (984), no. 3, [2] E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens. Michigan Math. J. 5 (958)

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26 26 JIGUANG BAO, HAIGANG LI, AND LEI ZHANG School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 00875, China address: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 00875, China address: Department of Mathematics, University of Florida, 358 Little Hall P.O.Box 805, Gainesville FL address: