Available olie at www.tjsa.com J. Noliear Sci. Appl. 9 (2016, 342 349 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 150001, 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. 2010 MSC: 35J40, 35J60. 1. 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 Email addresses: mcauchy@163.com (Xiayu Meg, fuyogqiag@hit.edu.c (Yogqiag Fu Received 2015-02-26
X. Y. Meg, Y. Q. Fu, J. Noliear Sci. Appl. 9 (2016, 342 349 343 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
X. Y. Meg, Y. Q. Fu, J. Noliear Sci. Appl. 9 (2016, 342 349 344 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].
X. Y. Meg, Y. Q. Fu, J. Noliear Sci. Appl. 9 (2016, 342 349 345 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
X. Y. Meg, Y. Q. Fu, J. Noliear Sci. Appl. 9 (2016, 342 349 346 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 + 1 + a 1 s 1 1 ds + s 1 + a 1 s R 1 2 x ds, x R. sds 1 ds C
X. Y. Meg, Y. Q. Fu, J. Noliear Sci. Appl. 9 (2016, 342 349 347 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.
X. Y. Meg, Y. Q. Fu, J. Noliear Sci. Appl. 9 (2016, 342 349 348 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. 11371110. Refereces [1] L. Caffarelli, Y. Y. Li, A extesio to a theorem of Jörges, Calabi ad Pogorelov, Comm. Pure Appl. Math., 56 (2003, 549 583. 2 [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, 261 301. 1, 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, 41 70. 1 [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, 1 67.3, 3, 3 [5] M. G. Cradall, P. L. Lios, Viscosity solutios of Hamilto-Jacobi equatios, Tra. Amer. Math. Soc., 277 (1983, 1 42. 2 [6] L. M. Dai, Existece of solutios with asymptotic behavior of exterior problems of Hessia equatios, Proc. Amer. Math. Soc., 139 (2011, 2853 2861. 1 [7] B. Gua, The Dirichlet problem for Hessia equatios o Riemaia maifolds, Calc. Var., 8 (1999, 45 69. 1 [8] B. Gua, J. Spruck, Hypersurfaces of costat curvature i hyperbolic space II, J. Eur. Math. Soc., 12 (2010, 797 817. 1 [9] B. Gua, J. Spruck, Szapiel M, Hypersurfaces of costat curvature i hyperbolic space I, J. Geometric Aal., 19 (2009, 772 795. 1 [10] H. Ishii, P. L. Lios, Viscosity solutios of fully oliear secod-order elliptic partial differetial equatios, J. Differetial Equatios, 83 (1990, 26 78. 3
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