2011 International Conference on Computer and Automation Engineering (ICCAE 2011)
IPCSIT vol. 44 (2012) © (2012) IACSIT Press, Singapore
DOI: 10.7763/IPCSIT.2012.V44.12
+
School of Mathematics and Information Science Weifang University
Weifang 261061, The People’s Republic of China
In this paper, we use the Perron method to prove the existence of viscosity solutions with asymptotic behavior at infinity to Hessian quotient equations.
viscosity solutions, asymptotic behaviour, Hessian quotient equations
In this paper, we study the Hessian quotient equation
S k , l
( D 2 u ) =
S k
( D 2 u
S l
( D 2 u )
)
= 1 in R n \ ∂ Ω . (1)
Here Ω ⊂⊂ R n is any strictly convex bounded domain, the function u , and
λ = ( λ
1
, " , λ n
) of D 2 u
S
0 ≤ l < k ≤ n ， D 2 u denotes the Hessian matrix of j
( D 2 u ) is defined to be the
, that is, j th elementary symmetric function of the eigenvalues
S j
( D 2 u )
=
= σ
1 ≤ j
( λ ( D 2 u ))
∑ i
1
< " < i j
λ
≤ i
1 n
" λ i j
, j = 1 , 2 , " , n .
When l = 0 , we denote S
0
( D 2 u ) ≡ 1 .
Hessian quotient equation (1) is an important class of fully nonlinear elliptic equation which is closely related to geometric problem. Some well-known equations can be regarded as its special cases. When l = 0 it is a k -Hessian equation. In particular, it is a Poisson equation if k = 1 equation if k = n . When k = n = 3 , l = 1 , that is, det( D 2 u ) = Δ u
, while it is a Monge-Amp è
, re
, (1) arises from special Lagrangian geometry([1]): if u is a solution of (1), the graph of Du over R 3 in C 3 is a special Lagrangian submanifold in
C 3 , that is, its mean curvature vanishes everywhere and the complex structure on C 3 sends the tangent space of the graph to the normal space at every point. Therefore (1) has drawn much attention, see [2-5].
The entire solutions of PDE have been studied by many authors, see [6,7]. In [CL], Caffarelli and Li have investigated the existence of solutions with asymptotic behavior to Monge-Amp è re equations in exterior domain. Dai in [D] has proved the existence of viscosity solutions with asymptotic behavior to
Hessian equations in exterior domain. In this paper, we study the viscosity solutions with asymptotic behavior at infinity to Hessian quotient equation (1) in R n \ ∂ Ω .
To work in the realm of elliptic equations, we have to restrict the class of functions. Let
Γ k
=
{
λ ∈ R n σ j
( λ ) > 0 , j = 1 , 2 , " , k
}
.
+
Corresponding author.
E-mail address : limeidai@yahoo.com.cn
65

λ =
A function
λ ( D 2 u ) = ( λ
1
, u
"
∈
, λ
C n
2
)
( R n \ ∂ Ω ) is called k -convex (uniformly k -convex) if λ is the eigenvalues of the Hessian matrix D 2 u .
∈ Γ k
( Γ k
)
From [2] and [3], we know that (1) is elliptic and
, where
( S k , l
1
( D 2 u )) k − l =
⎛
⎜⎜
S k
S l
(
( D 2 u )
D 2 u )
⎞
⎟⎟
1 k − l is a concave function of the second derivatives of u if u is uniformly k -convex. It is natural for the solutions of (1) to be considered in the class of uniformly k -convex functions.
An extensive study of viscosity solutions of second order partial differential equations can be found in [8] and [9].
For the reader's convenience, we recall the definition of viscosity solutions to Hessian quotient equations.
A function u satisfying
∈ C 0 ( R n \ ∂ Ω ) is called a viscosity subsolution of (1), if for any y ∈ R n \ ∂ Ω , ξ ∈ C 2 ( R n \ ∂ Ω ) u ( x ) ≤ ξ ( x ), x ∈ R n \ ∂ Ω , and u ( y ) = ξ ( y ) we have
S k , l
( D 2 ξ ( y )) ≥ 1 .
A function convex function u
ξ
∈
∈
C
C
0 ( R n
2 ( R n
\
\
∂ Ω )
∂ Ω )
is called a viscosity supersolution of (1), if for any
satisfying y ∈ R n \ ∂ Ω u ( x ) ≥ ξ ( x ), x ∈ R n \ ∂ Ω , and u ( y ) = ξ ( y )
, any k we have
S k , l
( D 2 ξ ( y )) ≤ 1 .
A function u ∈ C 0 ( R n \ ∂ Ω ) viscosity supersolution of (1).
is called a viscosity solution of (1), if u is both a viscosity subsolution and a
R n \
A function
∂ Ω , j u
= 1 , 2 , " , k
∈ C 0
.
( R n \ ∂ Ω ) is called k -convex if in the viscosity sense σ j
( λ ( D 2 u )) ≥ 0 in u ∈ C 0 convex.
( R n \ ∂ Ω ) is k -convex if and only if u is C 0 subharmonic; u is n -convex if and only if u is
From Proposition 2.2 in [9], we know the supremum of a set of subsolutions is still a subsolution.
Moreover, a comparison principle of viscosity solutions to Hessian quotient equations holds, see Theorem
3.3 in [8]. Then we can state the following existence and uniqueness results, see Proposition 2.3 in [9].
Lemma 1 Let B be a ball in R n and viscosity subsolution and supersolution of f ∈ C 0 ( B ) be nonnegative. Suppose u , u ∈ C 0 ( B ) are respectively
S k , l
( D 2 u ) = f in B , (2) and satisfy and u = ϕ u |
∂ B
= on ∂ B . u |
∂ B
= ϕ ∈ C 0 ( ∂ B ) , then there exists a unique k -convex function u ∈ C 0 ( B ) satisfying (2)
Lemma 2 Let viscosity sense S k , l
B be a ball in
( D 2 u ) ≥ f
R n and f ∈ C 0 ( B ) be nonnegative. Suppose in B . Then the Dirichlet problem u ∈ C 0 ( B ) satisfy in the
S k , l
( D 2 u ) u
= f in
= u on ∂ B
B , has a unique k -convex viscosity solution u ∈ C 0 ( B ) .
Lemma 3 Let D be an open set in v ∈ C 0 ( D ) , u ∈ C 0 ( R n
R
) satisfy respectively n and f ∈ C 0 ( R n ) be nonnegative. Assume k -convex functions
S k , l
( D 2 v ) ≥ f ( x ), x ∈ D ,
S k , l
( D 2 u ) ≥ f ( x ), x ∈ R n .
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Moreover, u ≤ u = v , x ∈ D , v , x ∈ ∂ D .
Set w ( x ) =
⎧ v
⎨
⎩ u
(
( x ), x ), x x
∈
∈
D ,
R n \ D .
Then w ∈ C 0 ( R n ) satisfies in the viscosity sense
S k , l
( D 2 w ) ≥ f ( x ), x ∈ R n .
Lemma 4 Let constant c
Ω ⊂ R n be a bounded strictly convex domain, ∂ only dependent of n , Ω , v
C 2 ( Ω )
such that for any ζ ∈ ∂ Ω
Ω ∈ C 2 , v ∈
there exists
C 2 ( Ω x ( ζ ) ∈
)
R
. Then there exists a n satisfying
| x ( ζ ) | ≤ c , w
ζ
( x ) < v ( x ), x ∈ Ω \ { ζ } , where w
ζ
( x ) = v ( ζ ) + c
*
2
( x − x ( ζ )
2 − ζ − x ( ζ )
2
,
) x ∈ R n , c
*
=
⎛
⎝
C l n
C n k
1
⎞
⎠ k − l
.
Our main result is the following theorem.
Theorem 1 exists a k
Let k − l ≥ 3 . For any c ∈ R
- convex viscosity solution u ∈ C 0
, there exists a constant β
0
∈ R
( R n \ ∂ Ω ) of (1) which satisfies
such that for any β > β
0
there lim
| sup x | → ∞
| x | k − l − 2 u ( x ) −
⎛
⎜⎜ c
*
2 x
2 + b ⋅ x + c
⎞
⎟⎟
< ∞ (4) u ( x ) = − β , x ∈ ∂ Ω , where c
*
=
⎛
⎝
C l n
C k n
⎞
⎠
1 k − l and Ω is any convex domain in R n .
Proof.
By subtracting a linear function from u , we only need to prove the theorem for b = 0 the proof into three steps.
. We divide
In the first step, we construct a viscosity subsolution of (1).
Let Ω be any bounded strictly convex domain in R n function satisfying
.Suppose Φ ∈ C 3 , α ( Ω ) (see [3]) is a k -convex
S k , l
( D 2 Φ ) = c
0
Φ
> 1 , x ∈ Ω ,
= 0 , x ∈ ∂ Ω .
By the comparison principle, Φ ≤ 0 such that in Ω . And by Lemma 4, for each ζ w
ζ
( x ) < Φ ( x ), x ∈ Ω \ { ζ } ,
∈ ∂ Ω , there exists x ( ζ ) ∈ R n
where w
ζ
( x ) = c
*
2
( x − x ( ζ )
2 − ζ − x ( ζ )
2
,
) x ∈ R n . and
ζ
∈ ∂ Ω x
.
Therefore w
ζ
( ζ ) = 0 , w
ζ
( x ) ≤ Φ ( x ) ≤ 0 , x ∈ Ω .
S k , l
( D 2 w
ζ
( x )) = 1 , x ∈ R n .
Thus w ( x ) = sup
ζ ∈ ∂ Ω w
ζ
( x ), x ∈ R n
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satisfies w ( x ) ≤ Φ ( x ), x ∈ Ω , (5) and from Proposition in [9],
S k , l
( D 2 w ) ≥ 1 , x ∈ R n .
Define
V ( x ) =
⎧ Φ
⎨
⎩ w (
( x ), x ), x x ∈
∈
Ω
R
, n \ Ω .
Then V ∈ C 0 ( R n ) . By (5) and Lemma 3, V satisfies in the viscosity sense
F ( D 2 V ) ≥ 1 , x ∈ R n .
Fix some R
1
> 0 such that
Ω ⊂⊂ B
R
1
, and let
R
2
= 2 R
1 c
*
.
For a > 1 , define w a
( x ) = inf
B
R 1
V + ∫
2 R
2 c
* x
( s k − l + a )
1 k − l ds , x ∈ R n .
A direct calculation gives for x > 0 , where y =
D ij w a
=
( y k − l + a k
) 1
− l
− 1
⎡
⎜
⎣
⎜
⎢
⎢
⎛
⎝ y k − l − 1 + a y c
*
δ ij
− ac
*
2 x i x j y
3
⎤
⎥
⎦
, c
* x . By rotating the coordinates, we may set x
D 2 w a
=
(
R k − l + a
) k
1
− l
− 1
⎜
⎝
⎜
⎜
⎛
⎜
R k − l − 1 c
*
0
"
0
R k − l
0
+
R
" a c
*
0
r
, therefore
"
"
"
"
0
0
R k − l
"
+ a
R c
*
⎟
⎠
⎟
⎟
⎞
⎟
, where R = y . So λ ( D 2 w a
) ∈ Γ k
.
Let R
0
= R k − l + a , then
S k , l
=
( D
R
0 k k
− l
2 w a
− k
R l k
0
l
− l
) =
⎢
⎣
⎡
C k n − 1
⎛
⎜⎜
⎢
⎢
⎡
⎣
C l n − 1
⎛
⎜⎜
S k
S l
(
( D 2
D w a
2 w a
)
)
R
0
R
R
0
R c
* c
*
⎞
⎟⎟ k
⎞
⎟⎟ l
+
+
R k − l − 1 c
*
C k n −
− 1
1
⎛
⎜⎜
R k − l − 1 c
*
C l − n −
1
1
⎛
⎜⎜
R
0 c
*
R
0
R
R c
*
⎞
⎟⎟
⎞
⎟⎟ k l − 1
− 1 ⎤
⎥
⎦
⎤
⎥
⎦
=
≥
R
0 c
* k − l R l − k
R
0 c
* k − l R l − k
C k n
R k − l
C l n
R k − l
+ aC k n − 1
+ aC l n − 1
C l n
C n k R k − l
R k − l + aC l n
= c k
*
− l
C n k
C l n
= 1 .
By the definition of R
2
, c
* x ≤
R
2
2
, x ≤ R
1
.
Consequently w a
( x ) ≤ inf
B
R
1
V
< inf
B
R 1
V
+ ∫
2
R
2
2
R
2
( s k − l + a k
) 1
− l
≤ V ( x ), x ≤ R
1
.
ds
(6)
68

Fix some R
3
> 3 R
2
satisfying
R
3 c
*
> 3 R
2
.
We choose a
1
> 1 such that for a ≥ a
1
, w a
( x ) > inf
B
R
1
V
≥ V ( x ),
+ ∫
2
3
R
2
R
2
( s k − l + a k
) 1
− l x = R
3
.
ds
Then by (6),
R
3
≥ R
1
.
By the definition of w a
, w a
( x ) = inf
B
R
1
V
= inf
B
R
1
V
= c
*
2
+
+
| x | 2
− c − 2 R 2
2
∫
2 R
2
∫
2 R
2 c
* x
+ c + inf
B
R
1
V
− c
* x
⎛
⎜ s
⎝
⎛
⎝
1 +
⎛
⎜ s
⎝
⎛
⎝
1 +
∫ ∞ c
* x
⎛
⎜ s
⎝
⎛
⎝
1 + a s k − l a s k − l
+ ∫
2
∞
R
2 s k a
− l k
1
− l k
1
− l
⎛
⎜ s
⎝
⎛
⎝
1 +
− 1
⎞
⎟
⎠ ds
− 1
⎞
⎟
⎠ ds + s a k − l
+ ∫
2 R
2 c
* x c
*
2 k
1
− l
| x sds
| 2
− 1
⎠
⎞
⎟ ds
− 2 R 2
2 k
1
− l
− 1
⎞
⎟
⎠ ds , x ∈ R n .
Let
μ ( a ) = inf
B
R
1
V + ∫ ∞
2 R
2
⎛
⎜ s
⎝
⎛
⎝
1 + s a k − l
⎞
⎠ k
1
− l
− 1
⎞
⎟
⎠ ds − c − 2 R 2
2
.
Then μ ( a ) is continuous and monotonic increasing for a and when a → ∞ , μ ( a ) → ∞ . And, w a
( x ) − μ ( a ) ≤ c
*
2 x
2 + c , a ≥ a
1
, x ∈ R n .
Moreover, when
For a ≥ x a
1
, set β
0
=
→ ∞ ,
μ ( a ) w a
and define for any u a
( x )
( x ) = c
* x
=
β >
2
β
0
, max
{
V w a
( x )
2 + c
( x ),
− β ,
+ w a x
μ ( a )
(
≥ x )
}
+
−
R
3
.
O
β ,
( ) x ≤ l
.
(7)
R
3
,
Then u a
( x ) = − β , x ∈ ∂ Ω , and by (7), when x → ∞ , u a
( x ) = c
*
2
Choose a
2
≥ a
1 sufficiently large such that for
V ( x ) − β ≤ V ( x ) − a ≥
β
0 a
2
, x
2 + c + O
= V ( x ) − inf
B
R
1
V + c + 2 R 2
2
2 − k + l (8)
− ∫
2
∞
R
2 s
⎝
⎜⎜
⎛
⎜
⎝
⎛
⎜ 1 + a s k − l ⎠
⎟
⎞
1 k − l
− 1
⎞
⎟
⎠
⎟⎟ ds
≤ c ≤ c
*
2 x
2 + c , x ≤ R
3
.
Therefore
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u a
( x ) ≤ c
*
2 x
2 + c , a ≥ a
2
, x ∈ R n .
By Lemma 3, u a
( x ) ∈ C 0 ( R n ) satisfies in the viscosity sense
S k , l
( D 2 u a
) ≥ 1 , x ∈ R n .
In the second step, we define the Perron solution of (1).
For a ≥ a
2
, let S a
denote the set of functions v ∈ C 0 ( R n ) satisfying
S k , l
( D 2 v ) ≥ 1 , x ∈ R n , v ( x ) = − β , x ∈ ∂ Ω , v ( x ) ≤ c
*
2 x
2 + c , x ∈ R n .
Clearly, u a
∈ S a
. Hence S a
≠ φ .Define in R n , u a
( x ) = sup
{ v ( x ) v ∈ S a
}
.
By the definition of u a
, u a
is a viscosity subsolution of (1) and satisfies u a
( x ) = − β , x ∈ ∂ Ω , u a
( x ) ≤ c
*
2 x
2 + c , x ∈ R n .
In the following, we prove u a
is a viscosity supersolution of (1).
In the third step, we prove u a
For any x
0
∈ R n \ ∂ Ω , ε > 0
is a viscosity solution of (1) satisfying (4).
, choose a ball B = B
ε
( x
0
) ⊂ R n \ ∂ Ω . By Lemma 2, the Dirichlet problem
S k , l
( D 2 u ) = 1 , x ∈ B , has a viscosity solution
~
∈ C 0 ( B )
= u a
, x ∈ ∂ B
. From the comparison principle, u a
ψ ( x ) =
⎧
~
⎨
⎩ u a
(
( x ), x ), x x
∈
∈
B ,
R n \ ( ∂ Ω ∪ B ).
≤
~
, x ∈ B . Define
By Lemma 3,
S k , l
( D 2 ψ ( x )) ≥ 1 , x ∈ R n \ ∂ Ω .
Because
S k , l
( D 2 ) = 1 =
= u a
≤
S k , l
( D 2 g ), x ∈ B , g , x ∈ ∂ B . where g ( x ) = c
*
2 x
2 + c , from the comparison principle,
≤ g , x ∈ B .
Hence ψ ∈ S a
. By the definition of u a
, u a
≥ ψ in R n . Consequently u ≤ u a
in B . As a result,
= u a
, x ∈ B .
Because x
0
is arbitrary, we know u a
is a viscosity supersolution of (1).
By the definition of u a
, u a
≤ u a
≤ g , x ∈ R n , so from (8), u a
satisfies (4). Theorem 1 is proved
The research was supported by Shandong Province Young and Middle-Aged Scientists Research Awards
Fund(BS2011SF025), Shandong Province Science and Technology Development Project(2011YD16002),
Shandong Province Natural Science Foundation(ZR2011AL008).
70

[1] R. Harvey and H. B. Lawson. Calibrated geometries, Acta Math . 1982, 148 : 47-157.
[2] L. Caffarelli , L. Nirenberg L and J. Spruck. The Dirichlet problem for nonlinear second-order elliptic quations.III.
Functions of the eigenvalues of the Hessian. Acta Math . 1985, 155 : 261-301.
[3] N. S. Trudinger, On the Dirichlet problem for Hessian equations. Acta Math . 1995, 175 : 151-164.
[4] J. G. Bao, J. Y. Chen, B. Guan, etal. Liouville property and regularity of a Hessian quotient equation. Am. J. math .
2003, 125 : 301-316.
[5] S. M. Liu and J. G. Bao. The local regularity for strong solutions of the Hessian quotient equation.
J. Math. Anal.
Appl . 2005, 303 : 462-476.
[6] L. Caffarelli L and Y. Y. Li. An extension to a theorem of Jörgens, Calabi, and Pogorelov. Comm. Pure Appl.
Math . 2003, 56 : 549-58.
[7] L. M. Dai and J. G. Bao. Multi-valued solutions to fully nonlinear uniformly elliptic equations. J. Math. Anal.
Appl.
2012, 389 : 314-321.
[8] M. G. Crandall , H. Ishii and P. L. Lions. User's guide to viscosity solutions of second order partial differential equations. B. Am. Math .Soc
. 1992, 27 : 1-67.
[9] H. Ishii. On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. Comm.
Pure Appl. Math.
1989, 42 : 15-45.
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