Existence of viscosity solutions with asymptotic behavior of exterior problems for Hessian equations

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1 Available olie at J. Noliear Sci. Appl. 9 (2016, Research Article Existece of viscosity solutios with asymptotic behavior of exterior problems for Hessia equatios Xiayu Meg, Yogqiag Fu Departmet of Mathematics, Harbi Istitute of Techology, Harbi , P. R. Chia. Abstract The Perro method is used to establish the existece of viscosity solutios of exterior problems for a class of Hessia type equatios with prescribed behavior at ifiity. c 2016 All rights reserved. Keywords: Hessia equatio, viscosity solutio, asymptotic behavior, exterior problem coicidece MSC: 35J40, 35J Itroductio I this paper, we study the Hessia equatio F (λ(d 2 u = σ > 0 x R \ Ω, (1.1 u = β x Ω, (1.2 where σ is a costat, Ω R ( 3 is a bouded domai, β is a costat, λ(d 2 u = (λ 1, λ 2,, λ are eigevalues of the Hessia matrix D 2 u. F is assumed to be defied i the symmetric ope covex coe Γ, with vertex at the origi, cotaiig Γ + = {λ R : each compoet of λ, λ i > 0, i = 1, 2,, }, ad satisfies the fudametal structure coditios: F i (λ = F λ i > 0 i Γ, 1 i (1.3 Correspodig author addresses: mcauchy@163.com (Xiayu Meg, fuyogqiag@hit.edu.c (Yogqiag Fu Received

2 X. Y. Meg, Y. Q. Fu, J. Noliear Sci. Appl. 9 (2016, ad F is a cotiuous cocave fuctio. I additio, F will be assumed to satisfy some more techical assumptios, such as F > 0 i Γ, F = 0 o Γ (1.4 ad for ay r 1, R > 0 1 F (R(, r,, r F (R(1, 1,, 1. (1.5 r 1 For every C > 0 ad every compact set K i Γ there is Λ = Λ(C, K such that F (Λλ C for all λ K. (1.6 There exists a umber Λ sufficietly large such that at every poit x Ω, if x 1,, x 1 represet the pricipal curvatures of Ω, the (x 1,, x 1, Λ Γ. (1.7 It is easy to verify that Γ {λ R : λ i > 0}. i=1 Equatio (1.5 is satisfied by each kth root of elemetary symmetric fuctio (1 k ad the k lth root of each quotiet of kth elemetary symmetric fuctio ad lth elemetary symmetric fuctio (1 l < k. The Hessia equatio (1.1 is a importat class of fully oliear elliptic equatios. There exist may excellet results i the case of bouded domais, see for examples [2, 3, 7, 13, 16] ad the refereces therei. Caffarelli, Nireberg ad Spruck [2, 3] ad Trudiger [15] established the classical solvability of the Dirichlet problems uder various hypothesis. I [11] Ivochkia, Trudiger ad Wag provided a simple approach the estimatio of secod derivatives of solutios. I [7] Gua studied the Dirichlet problems i bouded domais of Riemaia maifolds. Other boudary value problems have also bee cosidered. I [13], Trudiger treated the Dirichlet ad Neuma problems i balls for the degeerate case ad i [16] Urbas studied oliear oblique boudary value problems i two dimesios. But for ubouded domais there are few results i this directios. The study o this kid of fully oliear elliptic equatios is close to the ivestigatio o prescribed curvature equatios ad hypersurfaces of costat curvature with boudary, see for example [8, 9, 14] ad the refereces therei. Whe F (λ(d 2 u = σ k (λ(d 2 u, Γ = Γ k = {λ R : σ j > 0, j = 1, 2,, k}, where the kth elemetary symmetric fuctio σ k (λ = i 1 < <i k λ i1 λ ik for λ = (λ 1,, λ, i [6] Dai obtaied the followig result: Theorem 1.1. Let k 3. The for ay C R, there exists a costat β 0 R such that for ay β > β 0 there exists a k covex viscosity solutio u C 0 (R \ Ω of σ k (λ(d 2 u = 1, x R \ Ω satisfyig ( 1 where C = 1 k. C k ( ( lim sup x k 2 C u(x x 2 x 2 + C <, u = β x Ω

3 X. Y. Meg, Y. Q. Fu, J. Noliear Sci. Appl. 9 (2016, The followig theorem, which is the mai result i this paper, is a geeralizatio of Theorem 1.1 for k-hessia equatios. Theorem 1.2. Let k 3. The for ay C R, there exists a costat β 0 R such that for ay β > β 0 there exists a admissible viscosity solutio u C 0 (R \ Ω of (1.1 satisfyig ( ( R lim sup x 2 u(x x 2 x 2 + C <, u = β x Ω, where R is the costat satisfyig F (R, R,, R = σ. This paper is arraged as follows. I sectio 2, we give some prelimiary facts which will be used later. I sectio 3, we prove the mai result of this paper. 2. Prelimiaries The otio of viscosity solutios was itroduced by Cradall ad Lios [5]. Now viscosity solutio is a rather stadard cocept i partial differetial equatios. For the completeess of this paper, we first recall the otio of viscosity solutios. Defiitio 2.1. A fuctio u C 2 (R \ Ω is called admissible if λ(d 2 u Γ for every x R \ Ω. Defiitio 2.2. A fuctio u C 0 (R \ Ω is called a viscosity subsolutio(supersolutio to (1.1, if for ay y R \ Ω ad ay admissible fuctio ξ C 2 (R \ Ω satisfyig u(x ( ξ(x, x R \ Ω, u(y = ξ(y, we have F (λ(d 2 ξ(y ( σ. Defiitio 2.3. A fuctio u C 0 (R \ Ω is called a viscosity solutio to (1.1 if it is both a viscosity subsolutio ad a viscosity supersolutio to (1.1. Defiitio 2.4. A fuctio u C 0 (R \ Ω is called a viscosity subsolutio (supersolutio, solutio to (1.1-(1.2, if u is a viscosity subsolutio (supersolutio, solutio to (1.1 ad u (, =ϕ(x o Ω. Defiitio 2.5. A fuctio u C 0 (R \ Ω is called admissible if for ay y R \ Ω ad ay fuctio ξ C 2 (R \ Ω satisfyig u(x ( ξ(x, x R \ Ω, u(y = ξ(y, we have λ(d 2 ξ(y Γ. It is obvious that if u is a viscosity subsolutio, the u is admissible. Lemma 2.6. Let Ω be a bouded strictly covex domai i R, Ω C 2, ϕ C 2 (Ω. The there exists a costat C oly depedet o, ϕ ad Ω such that for ay ξ Ω, there exists x(ξ R such that x(ξ C, w ξ (x < ϕ(x for x Ω \ {ξ}, where w ξ (x = ϕ(ξ+ R 2 ( x x(ξ 2 ξ x(ξ 2 for x R ad R is the costat satisfyig F (R, R,, R = σ. This is a modificatio of Lemma 5.1 i [1].

4 X. Y. Meg, Y. Q. Fu, J. Noliear Sci. Appl. 9 (2016, Lemma 2.7. Let Ω be a domai i R ad f C 0 (R be oegative. fuctios v C 0 (Ω, u C 0 (R satisfy, respectively, Assume that the admissible Moreover Set F (λ(d 2 v f(x x Ω, F (λ(d 2 u f(x x R. w(x = u v, x Ω; u = v, x Ω. { v(x x Ω, u(x x R \ Ω. The w C 0 (R is a admissible fuctio ad satisfies i the viscosity sese F (λ(d 2 w(x f(x, x R. Lemma 2.8. Let B be a Ball i R ad f C 0,α (B be positive. Suppose that u C 0 (B satisfies i the viscosity sese F (λ(d 2 u f(x, x B. The the Dirichlet problem F (λ(d 2 u = f(x x B. u = u(x admits a uique admissible viscosity solutio u C 0 (B. We refer to [12] for the proof of Lemma 2.7 ad 2.8. x B 3. Proof of Mai Result We divide the proof of Theorem 1.2 ito two steps. Step 1. By [2], there is a admissible solutio Φ C (Ω of the Dirichlet problem: F (λ(d 2 Φ = C 0 > σ, x Ω, Φ = 0, x Ω. By the compariso priciples i [4], Φ 0 i Ω. Further by Lemma 2.6, for each ξ Ω, there exists x(ξ R such that W ξ (x < Φ(x, x Ω \ {ξ}, where ad sup x(ξ <. Therefore ξ Ω W ξ (x = R 2 ( x x(ξ 2 ξ x(ξ 2, ξ R W ξ (ξ = 0, W ξ (x Φ(x 0, x Ω, F (λ(d 2 W ξ (x = F (R, R,, R = σ, ξ R. Deote W (x = sup W ξ (x. ξ Ω

5 X. Y. Meg, Y. Q. Fu, J. Noliear Sci. Appl. 9 (2016, The ad by [10] Defie W (x Φ(x, x Ω, F (λ(d 2 W σ, x R. V (x = { Φ(x, x Ω, W (x, x R \ Ω. The V C 0 (R is a admissible viscosity solutio of F (λ(d 2 V σ, x R. Fix some R 1 > 0 such that Ω B R1 (0 where B R1 (0 is the ball cetered at the origi with radius R 1. Let R 2 = 2R 1 R 1 2. For a > 1, defie 1 R 2 x W a (x = if V + B R1 (s + a 1 ds, x R. The [ ( D ij W a = ( y + a 1 1 y 1 + a ] Rδ ij ar2 x i x j y y 3, x > 0, where y = R 1 2 x. By rotatig the coordiates we may set x = (r, 0, 0. Therefore ( D 2 W a = (R + a 1 1 Rdiag R 1, R 1 + a R,, R 1 + a, R where R = y. Cosequetly λ(d 2 W a Γ for x > 0 ad by (1.5 F (λ(d 2 W a F (R, R,, R = σ, x > 0. Moreover Fix some R 3 > 3R 2 satisfyig We choose a 1 > 1 such that for a a 1, W a (x V (x, x R 1. (3.1 R 3 R 1 2 > 3R 2. 3R2 W a (x > if V + B R1 (s + a 1 ds V (x, x = R3. The by (3.1, R 3 R 1. Accordig to the defiitio of W a, W a (x = if B R1 V + R 1 2 x = R 2 x 2 + C + if V + B R1 + 2R2 2 s R 1 2 x s 1 + a 1 s a 1 s 1 1 ds + s 1 + a 1 s R 1 2 x ds, x R. sds 1 ds C

6 X. Y. Meg, Y. Q. Fu, J. Noliear Sci. Appl. 9 (2016, Let µ(a = if B R1 V + + s 1 + a 1 s 1 ds C 2R2. 2 The µ(a is cotiuous ad mootoic icreasig for a ad whe a, µ(a. Moreover, W a (x = R 2 x 2 + C + µ(a O( x 2, whe x. (3.2 Defie, for a a 1, set β 0 = µ(a ad defie, for ay β > β 0, { max{v (x, Wa (x} β, x R u a (x = 3, W a β, x R 3. The by (3.2, u a (x = R 2 x 2 + C O( x 2, whe x ad by the defiitio of V, u a (x = β, Choose a 2 a 1 large eough such that whe a a 2, x Ω. V (x β V (x β 0 = V (x if B R1 V C + R 2 x 2 + C, x R 3. s 1 + a 1 s 1 ds + C + 2R2 2 Therefore u a (x R 2 x 2 + C, a a 2, x R. By Lemma 2.7, u a C 0 (R is admissible ad satisfies i the viscosity sese F (λ(d 2 u a σ, x R. Step 2. We defie the solutio of (1.1 by Perro method. For a a 2, let S a deote the set of admissible fuctio V C 0 (R which satisfies F (λ(d 2 V σ, V (x = β, x R \ Ω, x Ω, It is obvious that u a S a. Hece S a. Defie V (x R 2 x 2 + C, x R. u a (x = sup{v (x : V S a }, x R. Next we prove that u a is a viscosity solutio of (1.1. From the defiitio of u a, it is a viscosity subsolutio of (1.1 ad satisfies u a (x R 2 x 2 + C, x R.

7 X. Y. Meg, Y. Q. Fu, J. Noliear Sci. Appl. 9 (2016, So we eed oly to prove that u a is a viscosity supersolutio of (1.1 satisfyig (1.2. For ay x 0 R \ Ω, fix ε > 0 such that B = B ε (x 0 R \ Ω. The by Lemma 2.8, there exists a admissible viscosity solutio ũ C 0 (B to the pricelet problem By the compariso priciple i [4], Defie By Lemma 2.7, As where g(x = R 2 x 2 + C, we have F (λ(d 2 ũ = σ, x B, ũ = u a, x B. u a ũ, x B. (3.3 { ũ(x, x B, ψ(x = u a (x, x R \ {B Ω}. F (λ(d 2 ψ(x σ, x R. F (λ(d 2 ũ = σ = F (λ(d 2 g, x B, ũ = u a g, x B, ũ g, x B, by the compariso priciple i [4]. Therefore ψ S a. By the defiitio of u a, u a ψ i R. Cosequetly ũ u a i B ad further ũ = u a, x B i view of (3.3. Sice x 0 is arbitrary, we coclude that u a is a admissible viscosity solutio of (1.1. By the defiitio of u a, u a u a g, x R, so u a satisfies (1.2 ad we complete the proof of Theorem 1.2. Ackowledgemets: This work was supported by the Natioal Natural Sciece Foudatio of Chia (Grat No Refereces [1] L. Caffarelli, Y. Y. Li, A extesio to a theorem of Jörges, Calabi ad Pogorelov, Comm. Pure Appl. Math., 56 (2003, [2] L. Caffarelli, L. Nireberg, J. Spruck, The Dirichlet problem for oliear secod-order elliptic equatios III: Fuctios of eigevalues of the Hessias, Acta Math., 155 (1985, , 3 [3] L. Caffarelli, L. Nireberg, J. Spruck, Noliear secod order elliptic equatios V. The Dirichlet problem for Weigarte hypersurfaces, Comm. Pure Appl. Math., 4 (1988, [4] M. G. Cradall, H. Ishii, P. L. Lios, User s guide to viscosity solutios of secod order partial differetial equatios, Bull. Amer. Math. Soc., 27 (1992, , 3, 3 [5] M. G. Cradall, P. L. Lios, Viscosity solutios of Hamilto-Jacobi equatios, Tra. Amer. Math. Soc., 277 (1983, [6] L. M. Dai, Existece of solutios with asymptotic behavior of exterior problems of Hessia equatios, Proc. Amer. Math. Soc., 139 (2011, [7] B. Gua, The Dirichlet problem for Hessia equatios o Riemaia maifolds, Calc. Var., 8 (1999, [8] B. Gua, J. Spruck, Hypersurfaces of costat curvature i hyperbolic space II, J. Eur. Math. Soc., 12 (2010, [9] B. Gua, J. Spruck, Szapiel M, Hypersurfaces of costat curvature i hyperbolic space I, J. Geometric Aal., 19 (2009, [10] H. Ishii, P. L. Lios, Viscosity solutios of fully oliear secod-order elliptic partial differetial equatios, J. Differetial Equatios, 83 (1990,

8 X. Y. Meg, Y. Q. Fu, J. Noliear Sci. Appl. 9 (2016, [11] N. Ivochkia, N. Trudiger, X. J. Wag, The Dirichlet Problem for Degeerate Hessia Equatios, Comm. Partial Differetial Equatios, 29 (2004, [12] B. Tia, Y. Fu, Existece of viscosity solutios for Hessia equatios i exterior domais, Frot. Math. Chia, 9, (2014, [13] N. S. Trudiger, O degeerate fully oliear elliptic equatios i balls, Bull. Austral. Math. Soc., 35 (1987, [14] N. S. Trudiger, The Dirichlet problem for the prescribed curvature equatios, Arch. Ratioal Mech. Aal., 111 (1990, [15] N. S. Trudiger, O the Dirichlet problem for Hessia equatios, Acta Math., 175 (1995, [16] J. Urbas, Noliear oblique boudary value problems for Hessia equatios i two dimesios, A. Ist. H. Poicaré Aal. No liéaire, 12 (1995,