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2 J. Math. Aal. Appl. 45 (23) Cotets lists available at SciVerse ScieceDirect Joural of Mathematical Aalysis ad Applicatios joural homepage: O the exterior Dirichlet problem for Moge Ampère equatios Hogjie J,, Jiguag Bao b a School of Sciece, Beijig Uiversity of Posts ad Telecommuicatios, Beijig 876, PR Chia b School of Mathematical Scieces, Beijig Normal Uiversity, Laboratory of Mathematics ad Complex Systems, Miistry of Educatio, Beijig 875, PR Chia a r t i c l e i f o a b s t r a c t Article history: Received 27 October 22 Available olie 3 April 23 Submitted by Crista Gutierrez Keywords: Exterior Dirichlet problem Moge Ampère equatio Asymptotic behavior I this paper, we use the Perro method to prove the existece of viscosity solutios to a class of Moge Ampère equatios o exterior domais i R ( 2) with prescribed asymptotic behavior at ifiity. This exteds the Caffarelli Li s result o det(d 2 u) =. We also obtai the existece of etire covex viscosity solutios to the Moge Ampère equatios with prescribed asymptotic behavior at ifiity. 23 Elsevier Ic. All rights reserved.. Itroductio ad mai results I this paper, we will study the exterior Dirichlet problem for the Moge Ampère equatios det D 2 u = g(x), where g C (R ) is a positive fuctio. For the special case g, (.) is reduced to the equatio (.) det D 2 u =, which is well uderstood. A classical theorem of Jörges [3] for = 2, Calabi [6] for 5 ad Pogorelov [6] for all dimesios, respectively, states that ay classical covex etire solutio of (.2) must be a quadratic polyomial. Cheg ad Yau [7] gave a simpler ad more aalytical proof of the theorem. Aother proof of this theorem was give by Jost ad Xi i [4]. Caffarelli [4] exteded the Jörges Calabi Pogorelov theorem of classical solutios to viscosity solutios. It was proved by Trudiger ad Wag i [7] that the oly ope covex subset Ω of R which admits a covex solutio of det D 2 u = i Ω with lim x Ω u(x) = is Ω = R. I [5], Caffarelli ad Li ivestigated the existece of solutios to (.2) with prescribed asymptotic behavior at ifiity i exterior domais of R ( 2). The similar exterior Dirichlet problem of (.2) i R 2 was studied by Ferrer, Martíez, ad Milá i [8,9] usig complex variable methods. Bao ad Li [] also cosidered the exterior problem of (.2) for = 2 with differet asymptotic behaviors at ifiity. Recetly, Bao, Li ad Zhag [2] exteded Caffarelli Li s results i [5] to the case that g(x) = + O( x β ) at ifiity for some β > 2. (.2) Correspodig author. address: hjju@bupt.edu.c (H. Ju) X/$ see frot matter 23 Elsevier Ic. All rights reserved.

3 476 H. Ju, J. Bao / J. Math. Aal. Appl. 45 (23) Especially, for g(x) = ( + x 2 ) +2 α 2, the Moge Ampère equatio det D 2 u = ( + x 2 ) +2 α 2 (.3) comes from the study of traslatig solutios to the K α -flow, i.e., the flow by α powers of Gauss curvature. A more detailed backgroud of Eq. (.3) ca be foud i [5], i which the authors ad Jia studied the exterior Dirichlet problem of (.3). Motivated by [5,2,5], we study the exterior Dirichlet problem of the Moge Ampère equatio (.) with more geeral prescribed asymptotic behavior at ifiity, where g satisfies the followig assumptio: (A) g C (R ) is a positive fuctio with g(x) = g ( x ) + O( x β ) as x, where g C ([, + )) ad β >. Oe of the mai results i this paper is the followig theorem. Theorem.. Let D be a bouded, strictly covex ope subset of R ( 2) with D C 2 ad φ C 2 ( D). Assume that g satisfies (A) with b r α g (r) b 2 r α, r r for some positive costats b, b 2, r. If (mi{β, } 2) < α <, the for ay give b R, there exists some costat c, depedig oly o, b, b, b 2, α, β, D ad φ such that for every c > c there exists a locally covex viscosity solutio u C (R \ D) to the Dirichlet problem det D 2 u = g(x), i R \ D, (.5) u = φ, o D. Moreover, u satisfies lim sup x mi{β,} 2+α α u(x) f ( x ) b x c <, (.6) x where f ( x ) is the radially symmetric solutio of det D 2 u = g ( x ) i R with f () = f () =, give explicitly by (2.2). Remark.. We ca obtai u C (R \D) C (R \ D) by the regularity theory of the Moge Ampère equatio, if g C ; see [,3]. Remark.2. It is ecessary for the coditio (.4) by couterexample i the last sectio. We ca see that c > c is also ecessary from Wag ad Bao s results i [8], i which they obtaied the ecessary ad sufficiet coditios o existece ad covexity of radial solutios for (.2) o exterior domais. Remark.3. Our result is compatible with Theorem.5 i [5] for g =, ad Theorem. i [2] for g =, respectively. Remark.4. There are may issues to be solved, oe of which is to weake the boudary coditio φ C 2 ( D) to φ C ( D). Besides, the Dirichlet problems o the geeral ubouded domais or half space are also iterestig problems. I this paper, we also obtai the existece of etire covex viscosity solutio to the Moge Ampère equatio (.) with give asymptotic behavior at ifiity. Theorem.2. Assume g, g ad f are as i Theorem.. For ay give b R, there exists some costat c, depedig oly o, b, b, b 2, α ad β, such that for every c > c there exists a etire covex viscosity solutio u C (R ) to Eq. (.). Moreover, u satisfies (.6). The paper is orgaized as follows. I Sectio 2, we fid the radially symmetric solutio to det D 2 u = g by the ODE method ad show the asymptotic behavior of the solutio at ifiity for α (, ). Theorems. ad.2 will be proved by the Perro method i Sectios 3 ad 4, respectively. Fially, we give a couterexample to show the ecessity of the coditio (.4) i Sectio 5. (.4) 2. Radially symmetric solutios of det D 2 u = g Let u(x) = f ( x ). The det(d 2 f (r) u(x)) = f (r), r r = x.

4 H. Ju, J. Bao / J. Math. Aal. Appl. 45 (23) The equatio det D 2 u(x) = g ( x ) ca be rewritte as (f (r)) f (r) = r g (r). (2.) By the similar method as i [5], we ca fid a covex radially symmetric solutio u(x) := f ( x ) C 2 (R ) if we defie f () = g (). Deote by f (r) the solutio of (2.) with f () =, f () =. The f (r) = r τ s g (s)ds dτ. (2.2) Assume that b r α g (r) b 2 r α, for r > r (2.3) for some fixed costat r. Sice τ r τ s g (s)ds = s g (s)ds + s g (s)ds, the, we obtai from (2.3) that b τ + α τ +α + m r s g (s)ds b 2 + α τ +α + m 2, for τ >, (2.4) where m i (i =, 2) deped o r,, α ad b i (i =, 2). Therefore, f (r) = O r 2+ α, as r. 3. Proof of Theorem. By subtractig a liear fuctio from u, we eed oly to prove the theorem for b =. Uder the assumptio (A), we ca fid two positive cotiuous fuctios g ad ḡ satisfyig with < g( x ) g(x) ḡ( x ), x R g( x ) = g ( x ) c x β, ḡ( x ) = g ( x ) + c 2 x β, (3.) (3.2) (3.3) for some positive costats c ad c 2 ad x sufficietly large. By the similar techique as i [5, Lemma A.3], we obtai the followig existece theorem i bouded covex domais, which is eeded for us to use the Perro method. Lemma 3.. Let Ω be a smooth, bouded, strictly covex subset i R ( 2), ad let g be a positive cotiuous fuctio o Ω ad ϕ C ( Ω). Assume that u C ( Ω) is a covex viscosity subsolutio to det D 2 u = g(x). The the Dirichlet problem det D 2 u = g(x), i Ω, u = u, o Ω has a uique covex viscosity solutio u C ( Ω). Lemma 3.2. Let Ω Ω 2 be two bouded domais i R ad g C (R ) a oegative fuctio. Suppose that covex fuctios v C ( Ω 2 ), u C (R \ Ω ) satisfy det D 2 v g(x), x Ω 2, det D 2 u g(x), x R \ Ω i the viscosity sese, respectively, ad u < v, x Ω, u > v, x Ω 2.

5 478 H. Ju, J. Bao / J. Math. Aal. Appl. 45 (23) Set v(x), x Ω, w(x) := max{v(x), u(x)}, x Ω 2 \ Ω, u(x), x R \ Ω 2. The, w C (R ) is a covex fuctio ad satisfies det D 2 w g(x), x R, i the viscosity sese. Proof. Let x R, ϕ C 2 (R ) satisfy w( x) = ϕ( x), w(x) ϕ(x), x R. (3.4) If x Ω, we have v( x) = w( x) = ϕ( x), v(x) w(x) ϕ(x), x Ω 2. (3.5) By the defiitio of viscosity subsolutio, we obtai det D 2 ϕ( x) g( x). (3.6) If x R \ Ω 2, we have u( x) = w( x) = ϕ( x), u(x) w(x) ϕ(x), x R \ Ω. (3.7) The (3.6) holds, sice u is a viscosity subsolutio of det D 2 u = g(x) i R \ Ω. I the case of x Ω 2 \ Ω, if w( x) = v( x), the we have (3.5), which implies (3.6). If w( x) = u( x), the we obtai (3.7), which implies (3.6). The lemma is completed. Defiitio 3.3. The subfuctio class S c for some costat c is defied as follows: a fuctio v S c if ad oly if () v C (R \ D) ad v φ o D; (2) v is a locally covex viscosity subsolutio of (.) i R \ D; (3) v(x) w(x) + c, x R \ D, where w(x) := x τ s g(s)ds dτ. Lemma 3.4. There exists some costat c, depedig oly o, α, β, D ad φ, such that, for ay c > c, we ca fid a viscosity subsolutio u S c. Proof. Fix R 2 > R > such that D B R () ad R 2 > 3R. Let C := max x B R2 g(x) >. Accordig to Lemma 5. i [5], for ay boudary poit ξ D, we ca costruct a covex smooth solutio v ξ (x) to the equatio det D 2 u = C i R with v ξ (ξ) = φ(ξ), v ξ < φ o D \ {ξ}. Defie V(x) := sup v ξ (x), x B R2 (). ξ D We ca see that V(x) is a covex viscosity subsolutio of (.) i B R2 () ad satisfies V(ξ) φ(ξ), ξ D. O the other had, by the defiitio of V, for ay ξ D, V(ξ) v ξ (ξ) = φ(ξ).

6 H. Ju, J. Bao / J. Math. Aal. Appl. 45 (23) Therefore, V = φ o D. For a, we defie a fuctio x (x) := if V(x) + [ h(τ) + a] dτ, x B R 2R where h(τ) = τ s ḡ(s)ds. It is easy to check that is a locally covex subsolutio of (.) i R \ B () ad (x) V(x), for x R. (3.8) I view of R 2 > 3R, we choose a > large eough such that for a a, 3R (x) if V(x) + [ h(τ) + a] dτ + V(x), x = R 2. (3.9) x B R 2R Let h (τ) = τ s g (s)ds. The f (r) = r [h (τ)] dτ. Now, we rewrite wa as (x) = f ( x ) + if x B R V(x) + = f ( x ) + µ(a) x x [ h(τ) + a] dτ 2R (h (τ)) x [h (τ)] dτ + h(τ) h (τ) + a h (τ) dτ, (3.) where µ(a) := if V(x) f (2R ) + (h (τ)) x B R 2R + h(τ) h (τ) + a h (τ) dτ. It follows from (3.3) that h(τ) = h (τ) + = h (τ) + ĉ + ĉ 2 τ β, s (ḡ(s) g (s))ds + τ s (ḡ(s) g (s))ds as τ large eough, where ĉ i (i =, 2) deped o, c, c 2 ad β. It is easy to compute that α+β > sice α > (mi{β,} 2), which together with (2.4) ad (3.) implies that for τ sufficietly large, ĉ 3 τ mi{β,} α+ α (h (τ)) + h(τ) h (τ) + a h (τ) ĉ 4τ mi{β,} α+ α, where ĉ i (i = 3, 4) depeds o a, α, β,, c i (i =, 2). The, i view of (mi{β,} 2) < α <, i.e., mi{β, } α+ α <, we have (3.) µ(a) < (3.2) ad (x) = f ( x ) + µ(a) + O x 2 mi{β,} α+ α, as x. (3.3) As above, let τ h(τ) = s g(s)ds,

7 48 H. Ju, J. Bao / J. Math. Aal. Appl. 45 (23) the w(x) ca be rewritte as x x w(x) = f ( x ) + [h(τ)] dτ [h (τ)] dτ where = f ( x ) + µ µ := f () + x (h (τ)) (h (τ)) + h(τ) h (τ) + h(τ) h (τ) h (τ) h (τ) dτ. dτ, (3.4) By the similar calculatig as (3.) (3.3), we obtai µ < ad w(x) = f ( x ) + µ + O x 2 mi{β,} α+ α, as x. (3.5) I view of (3.), for a >, (h (τ)) + h(τ) h (τ) + a h (τ) (h (τ)) + h(τ) h (τ). h (τ) The, (x) w(x) + µ(a) µ, x R \ B (). (3.6) It is clear that µ(a) is cotiuous, mootoic icreasig for a, ad µ(a) as a. We ca choose a 2 > large eough such that, for a > a 2, V(x) w(x) + µ(a) µ, x R 2. (3.7) Set a = max{a, a 2 }. The for ay a > a, (3.9), (3.6) ad (3.7) hold. Defie V(x), x < R, (x) := max{v(x), (x)}, R x < R 2, (x), x R 2. (3.8) Obviously, (x) = V(x) = φ(x), x D. (3.9) By Lemma 3.2, is a covex viscosity subsolutio of (.) i R. For c > c := µ(a ), there is a umber a > a, such that c = µ(a). From (3.6) ad (3.7), we have, for c > c, (x) w(x) + c µ, x R. The S c µ. Moreover, by (3.3), (x) = f ( x ) + µ(a) + O x 2 mi{β,} α+ α, as x. (3.2) Takig c = c µ, the lemma is proved. Defie for c > c, u c (x) := sup{v(x) : v S c µ }, x R \ D. Lemma 3.5. The fuctio u c (x) C (R \ D) is a locally covex viscosity solutio to the exterior Dirichlet problem (.5) ad u c w(x) + c µ i R \ D. Proof. We will divide the proof ito two steps. The first step is to prove that u c (x) is a locally covex viscosity subsolutio of (.) with u c = φ o D ad u c w(x) + c µ i R \ D.

8 H. Ju, J. Bao / J. Math. Aal. Appl. 45 (23) As i [5], from the defiitio of u c ad S c µ, we kow that u c is a locally covex viscosity subsolutio ad u c w(x) + c µ i R \ D. Sice u c (x) (x) i R \ D with c = µ(a) ad is cotiuous o the boudary of D, so, for ay ξ D, lim if x ξ u c (x) (ξ ) = φ(ξ ). (3.2) O the other had, for ay v S c µ, v is a viscosity subsolutio of (.) i R \ D, i.e., for every x R \ D ad every fuctio ϕ C 2 (R \ D) satisfyig ϕ( x) = v( x), ϕ v o R \ D, we have det D 2 ϕ( x) g( x) >. By Remark.3.2 i [], we kow D 2 ϕ( x). The, ϕ( x) [det D 2 ϕ( x)] >, which meas that v is a viscosity subsolutio of v = i R \ D ad v φ o D. Choosig a ball B R () D. It is well-kow that the Dirichlet problem v + =, i B R () \ D, v + = φ, o D, v + = u c, o B R () (3.22) has a uique classical solutio v + C 2 (B R () \ D) C (B R () \ D); see []. Applyig a compariso priciple to v + ad v S c µ, v v + i B R () \ D. Therefore, u c v + i B R () \ D ad lim sup x ξ u c (x) v + (ξ ) = φ(ξ ). The secod step is to show that u c (x) is a viscosity solutio of (.) i R \ D. For ay x R \ D, choose a ball Bε (x ) R \ D. By Lemma 3., there is a uique covex viscosity solutio ũ C (B ε (x )) to the Dirichlet problem det D 2 ũ = g(x) i B ε (x ), ũ = u c o B ε (x ). Accordig to the defiitio of w(x) ad u c, we ca see that det D 2 ( w + c µ ) g(x), i B ε (x ), w + c µ u c o B ε (x ). Usig the compariso priciple of viscosity solutios, see the mai theorem i [2], we have ũ u c ad ũ w + c µ o B ε (x ). Defie ũ(x), x Bε (x w(x) = ), u c (x), x R \ (D B ε (x )). Obviously, w is a locally covex viscosity subsolutio with w = φ o D ad u c w+c µ i R \D; therefore, w S c µ. The, u c w o B ε (x ) by the defiitio of u c. It follows that u c ũ o B ε (x ), ad u c C (R \ D) is a viscosity solutio to (.5). Proof of Theorem.. From Lemma 3.5, we eed oly to prove that u c satisfies (.6). I fact, by the defiitio of u c ad Lemma 3.5, we have u c w(x) + c µ i R \ D, where c = µ(a), which, together with (3.5) ad (3.2), implies lim sup x mi{β,} 2+α α uc (x) f ( x ) c <. x The theorem is completed.

9 482 H. Ju, J. Bao / J. Math. Aal. Appl. 45 (23) Proof of Theorem.2 As above, we defie a class of subfuctios as follows. Defiitio 4.. The subfuctio class Ŝ c for some costat c is defied as follows: a fuctio v is i Ŝ c if ad oly if () v C (R ) is a covex viscosity subsolutio of (.) i R ; (2) v(x) w(x) + c, x R. From the proof of Lemma 3.4, we ca see that is a covex viscosity subsolutio of (.) i R, ad (x) w(x)+c µ with c = µ(a) ad a > a. Therefore, Ŝ c µ for c > c. Moreover, (x) = f ( x ) + µ(a) + O x 2 mi{β,} α+ α, as x. (4.) Lemma 4.2. If we defie û c (x) := sup{v(x) : v Ŝ c µ }, x R, c > c. The û c is a etire covex viscosity solutio of (.) with û c w(x) + c µ i R. Proof. The proof is similar to the proof of Lemma 3.5. Proof of Theorem.2. It follows from Lemma 4.2 that for ay c > c, there exists a etire covex viscosity solutio û c C (R ) to (.). We eed oly to prove (.6). By the defiitio of û c ad Lemma 4.2, we have û c w + c µ, i R, where c = µ(a). The the asymptotic behavior (.6) follows from (3.5) ad (4.). The theorem is completed. 5. Example I this part, we will give a couterexample to show the ecessity of α > (mi{β,} 2) i Theorem.. From Theorems 4.2 ad 4.3 i [5] where β =, we ca see that it is ecessary for α > ( 2) i Theorem.. Fix a ball B () B R () R ( 2) ad a costat d. We cosider the locally covex radially symmetric solutio of the exterior Dirichlet problem where with det D 2 u = g( x ) i R \ B R (), u = d, o B R (), g( x ) = g ( x ) + x β g ( x ) =, x, x α, x >. for x R Theorem 5.. Assume α = (β 2) for < β <. The ay radially symmetric locally covex solutio u(x) := f ( x ) C (R \ B R ()) C 2 (R \ B R ()) to the exterior problem (5.) has the followig asymptotic behavior at ifiity: lim sup x β u(x) f ( x ) c c 2 l x <, (5.2) x where c 2 =, ad c β +α depeds o d, R,, β ad f (R). Proof. Assume that u(x) := f ( x ) C (R \ B R ()) C 2 (R \ B R ()) is a locally covex radially symmetric solutio of (5.). The f (r) >, f (r) > for r > R, r = x ad r ((f (r)) ) = r g(r). Itegratig the above equatio o [R, r] for r > R, we obtai f (r) = r where b = (f (R)). R s g(s)ds + b, (5.)

10 H. Ju, J. Bao / J. Math. Aal. Appl. 45 (23) Let h(r) = r R s g(s)ds, h (r) = r s g (s)ds. Uder the assumptio that g(r) = g (r) + r β for r R, it is easy to see that h(r) = h (r) h (R) + r = h (r) c + β r β for r R, where c = h (R) + x R s β ds β R β. The, by recallig the defiitio of f, f ( x ) = [h(τ) + b] dτ + f (R) R x = f ( x ) f (R) + d + (h (τ)) β + τ β c + b h (τ) I view of α = (β 2) (h (τ)) + as τ. Therefore, R for < β <, by Taylor s expasio, we obtai β τ β c + b h (τ) = β + α + b c f ( x ) = f ( x ) + c + c 2 l x + O( x β ), x, where c 2 =, ad c β +α depeds oly o, β, b, d ad R. Ackowledgmets + α τ dτ. τ β + o(τ β ), The authors would like to thak the referee for very helpful commets ad suggestios. The first author was supported by the Natioal Natural Sciece Foudatio of Chia (2), the Fudametal Research Fuds for the Cetral Uiversities (23RC9) ad the Importatio ad Developmet of High-Caliber Talets Project of Beijig Muicipal Istitutios (CIT&TCD23429). The work of the secod author was supported by Program for Chagjiag Scholars ad Iovative Research Team i Uiversity (IRT98), the Natioal Natural Sciece Foudatio of Chia (72), ad Doctoral Program Foudatio of Istitutio of Higher Educatio of Chia (233). Refereces [] J.G. Bao, H.G. Li, O the exterior Dirichlet problem for the Moge Ampère equatio i dimesios two, Noliear Aal. 75 (8) (22) [2] J.G. Bao, H.G. Li, L. Zhag, Moge Ampère equatio o exterior domais. [3] L. Caffarelli, Iterior W 2,p estimates for solutios of the Moge Ampère equatio, A. of Math. (2) 3 () (99) [4] L. Caffarelli, Topics i PDEs: The Moge Ampère Equatio. Graduate Course, Courat Istitute, New York Uiversity, 995. [5] L. Caffarelli, Y.Y. Li, A extesio to a theorem of Jörges, Calabi ad Pogorelov, Comm. Pure Appl. Math. 56 (23) [6] E. Calabi, Improper affie hyperspheres of covex type ad a geeralizatio of a theorem by K. Jörges, Michiga Math. J. 5 (958) [7] S.Y. Cheg, S.T. Yau, Complete affie hypersurfaces, part I. The completeess of affie metrics, Comm. Pure Appl. Math. 39 (986) [8] L. Ferrer, A. Martíez, F. Milá, A extesio of a theorem by K. Jörges ad a maximum priciple at ifiity for parabolic affie spheres, Math. Z. 23 (3) (999) [9] L. Ferrer, A. Martíez, F. Milá, The space of parabolic affie spheres with fixed compact boudary, Moatsh. Math. 3 () (2) [] D. Gilbarg, N. Trudiger, Secod Order Elliptic Partial Differetial Equatios, secod ed., Spriger-Verlag, 983. [] C.E. Gutièrrez, The Moge Ampère Equatio, i: Progress i Noliear Differetial Equatios ad their Applicatios, vol. 44, Birkhäuser, Bosto, 2. [2] R. Jese, P.L. Lios, P.E. Sougaidis, A uiqueess result for viscosity solutios of secod-order fully oliear partial differetial equatios, Proc. Amer. Math. Sci. 2 (987) [3] K. Jörges, Über die Lösuge der differetialgleichug rt s 2 =, Math. A. 27 (954) [4] J. Jost, Y.L. Xi, Some aspects of the global geometry of etire space-like submaifolds, Results Math. 4 (2) [5] H.J. Ju, J.G. Bao, H.Y. Jia, Existece for traslatig solutios of Gauss curvature flow o exterior domais, Noliear Aal. 75 (8) (22) [6] A.V. Pogorelov, O the improper affie hyperspheres, Geom. Dedicata (972) [7] N.S. Trudiger, X.J. Wag, The Berstei problem for affie maximal hypersurfaces, Ivet. Math. 4 (2) (2) [8] C. Wag, J.G. Bao, Necessary ad sufficiet coditios o existece ad covexity of solutios for Dirichlet problems of Hessia equatios o exterior domais, Proc. Amer. Math. Soc. 4 (23)