The Dirichlet Problem of Hessian Equation in Exterior Domains

In this paper, we will obtain the existence of viscosity solutions to the exterior Dirichlet problem for Hessian equations with prescribed asymptotic behavior at infinity by the Perron’s method. This extends the Ju–Bao results on Monge–Ampère equations det D2u = f (x).


Introduction
In this paper, we shall study the exterior Dirichlet problem of Hessian equation where Ω ⊂ R n is a bounded, strictly convex set and 0 ∈ Ω, f ∈ C 0 (R n ) is a positive function. If k = 1, (1) is reduced to Poisson's equation ∆u = f (x). If k ≥ 2, (1) is fully nonlinear elliptic. When k = n, we can derive the Monge-Ampère equation det D 2 u = f (x). There are many results of interior Hessian equations, see [1][2][3][4][5][6][7] and the correlative literatures. For example, Caffarelli-Nirenberg-Spruck [2] obtained the existence result for the interior Dirichlet problem of Hessian equations.
Besides the interior Dirichlet problems, the exterior Dirichlet problem has also been extensively studied. The exterior Dirichlet problem is closely related to the classical theorem of Jörgens (n = 2 [8]), Calabi (n ≤ 5 [9]), and Pogorelov (n ≥ 2 [10]) which states that any classical convex solution of det D 2 u = 1 in R n must be a quadratic polynomial. A more simplified and analytical proof of the theorem was obtained by Cheng-Yau [11]. Jost-Xin [12] proved the theorem in different ways. Later, Caffarelli [13] generalized the conclusion to the viscosity solution. However Trudinger-Wang [14] showed that if u ∈ C 2 (Ω) is a convex function of det D 2 u = 1 and Ω is a convex set of R n with lim x→∂Ω u(x) = ∞, then u is quadratic and Ω = R n .
In 2003, Caffarelli-Li [15] proved the existence result to Monge-Ampère equation in R n (n ≥ 3). That is, let det D 2 u = 1, whereφ ∈ C 2 (∂Ω). Letb ∈ R n ,c ∈ R and A = {A is a real n × n symmetric positive definite matrice satisfying det A = 1.} There is some constant c * and A ∈ A, c * depends on n, Ω,φ,b, for everyc > c * , then the problem (2) has a function u ∈ C ∞ (R n \Ω) ∩ C 0 (R n \Ω) satisfying the asymptotic behavior lim sup If n = 2, by complex variable methods, Ferrer et al. [16,17] investigated the Dirichlet problem earlier. Then the exterior Dirichlet problem of Monge-Ampère equation was investigated by [18][19][20][21][22][23] and related literatures. For instance, in [20], Ju-Bao proved the existence result to det , β > 0 on exterior domains. Motivated by [15], the second author and Bao [24] first studied the Dirichlet problem of Hessian equation S k (D 2 u) = 1 on exterior domains. The existence result with the asymptotic behavior (3) was obtained for A = (C k n ) −1/k I and I is the identity matrix in [24]. More excellent achievements about the exterior problem of Hessian equations can be referred to [25][26][27]. Specially, Bao-Li-Li [25] extended the asymptotic behavior (3) to more general A and Cao-Bao [26] studied the exterior problem of Hessian equation In this paper, we'll generalize the outcome of Monge-Ampère equation in [20] to the Hessian equation (1). We will study the exterior Dirichlet problem In order to make the Hessian equation to be elliptic, we have to limit a class of functions. Set Let u ∈ C 2 (R n \Ω) and λ(D 2 u) represent the eigenvalues λ 1 , λ 2 , · · · , λ n of the Hessian matrix D 2 u. If λ ∈ Γ k , then we call u k-convex. Definition 1. [24] A function u ∈ C 0 (R n \ Ω) is known as a viscosity supersolution (respectively, subsolution) to S k (D 2 u) = f (x), if for any t ∈ R n \ Ω, ε ∈ C 2 (R n \ Ω) satisfying ε(t) = u(t) and ε(x) ≤ (respectively, ≥)u(x) on R n \ Ω, we get S k (D 2 ε(t)) ≤ (respectively, ≥) f (t).
In viscosity supersolution, we need ε(x) to be k−convex.
If u ∈ C 0 (R n \ Ω) is a viscosity supersolution, meanwhile, a viscosity subsolution, we call that u ∈ C 0 (R n \ Ω) is a viscosity solution to S k (D 2 u) = f (x).

Theorem 2.
Let f , f 0 , g 0 be as in Theorem 1. For any given b 0 ∈ R n , there is some constant m * , m * depends on b 0 , b 1 , b 2 , n, α, β, then for any m > m * , the equation has an entire k−convex solution u ∈ C 0 (R n ) in the viscosity sense. In addition, u satisfies (6).
This paper can be divided into the following sections. In the second part, we give the radially symmetric solution to S k (D 2 u) = f 0 . The third and fourth parts will prove Theorems 1 and 2, respectively. In the last section, we show the importance of condition (5) by counterexample.

Radially Symmetric Solutions of
We can choose x = (r, 0, · · · 0) T , then From (8), we can get and where R 1 is a positive number and Then the radially symmetric solution

Proof of Theorem 1
Known from [20], by subtracting a linear function from u, let us suppose that b 0 = 0 in (6). Set f and f be two positive functions and satisfy x ∈ R n for some positive numbers c 1 , c 2 and |x| large enough.
Lemma 1. Let D be a smooth, bounded subset of R n , f be a positive function on D and a function u Lemma 2. Let D 1 ⊂ D 2 be two smooth, bounded sets in R n and f ∈ C 0 (R n ) be positive. In the viscosity sense, Then in the viscosity sense, v ∈ C 0 (R n ) is a k−convex function and satisfies The proofs of Lemma 1 and Lemma 2 can be referred to [20,24], here we omit the proofs. For some constantm, let Sm be a set satisfying that a function w ∈ Sm if and only if Lemma 3. There is some constantm,m depends on Ω, n, ϕ, α, β, then for anym >m, there is a viscosity subsolution u ∈ Sm.
According to [15], for any ε ∈ ∂Ω, there exists a k−convex solution w ε (x) to the following equation Through the definition of W, for any ε ∈ ∂Ω, where When τ is large enough, we can get Moreover, from (13) and the above, we can see that where Similarly, we can get Since therefore Obviously, µ(b) for b is continuous, monotonic increasing, and µ(b) → ∞ as b → ∞. Fix b 2 > 0 large enough such that for b > b 2 , We know that . According to (20) and (21), we can know, for m > m * , Therefore, u b ∈ S m−µ 0 . In addition, according to (15) , we have Then the lemma can be proved withm = m * − µ 0 .
Set m > m * , define Lemma 4. The function u m ∈ C 0 (R n \ Ω) is a locally k−convex solution to (4) and u m ≤ v + m − µ 0 , x ∈ R n \ Ω, in the viscosity sense.
Proof. According to the definition of u m and S m−µ 0 , it is clear that u m is a locally k−convex viscosity subsolution to (1) and u m ≤ v + m − µ 0 in R n \ Ω. Firstly, we just have to show that u m = ϕ on ∂Ω. We can get by the proof of Lemma 3, Since u b is continuous on ∂Ω, then, for any ε 0 ∈ ∂Ω, we have Now we prove lim sup For any w ∈ S m−µ 0 , w is a viscosity subsolution in R n \ Ω, that is, for every t ∈ R n \ Ω and From the above equation, we can see that w is a viscosity subsolution of ∆w = 0 in R n \ Ω and w ≤ ϕ on ∂Ω.
Fix a ball B R (0) ⊃ Ω. For the Dirichlet problem Using the comparison principle ( [28,29]) for w + and w ∈ S m−µ 0 , we have From the definition of u m and v(x), then Applying the comparison principle to viscosity solutions, we haveû ≥ u m andû . By the definition of u m , we have u m ≥ŵ m on B λ (x 0 ). We can know that u m ≡û on B λ (x 0 ) and u m ∈ C 0 (R n \ Ω) is a k-convex viscosity solution of (4).
Proof of Theorem 1. According to the above, we just have to show that u m satisfies (6). According to Lemma 4 and the definition of u m , we have (17) and (23), we can know The theorem can be proved.

Proof of Theorem 2
For some constantm, letŜm be a set satisfying that a function w ∈Ŝm if and only if (2) w(x) ≤ v(x) +m, for any x ∈ R n .
According to Lemma 3, u b is a k−convex viscosity subsolution of (1), for m > m * . Therefore, Lemma 5. Define for m > m * , It is clear thatû m is a k−convex viscosity solution of S k (

Proof.
A similar method to prove this Lemma can be acquired in Lemma 4.
Proof of Theorem 2. Known from Lemma 5, for any m > m * ,û m ∈ C 0 (R n ) is a k−convex solution to (1). We need to prove (6). According to Lemma 5 andû m , then, (17) and (25), we can know Theorem 2 can be proved.

Example
In the last part, we demonstrate the importance of α > k(2 − min{n, β}) 1 − k by a counterexample.
We shall obtain a locally k−convex radially symmetric solution which satisfies and lim α, H, β, a, b, c.