Open Access

Existence of viscosity multi-valued solutions with asymptotic behavior for Hessian equations

Boundary Value Problems20142014:165

https://doi.org/10.1186/s13661-014-0165-8

Received: 5 March 2014

Accepted: 25 June 2014

Published: 24 September 2014

Abstract

The Perron method is used to establish the existence of viscosity multi-valued solutions for a class of Hessian-type equations with prescribed behavior at infinity.

Keywords

Hessian equation multi-valued solution asymptotic behavior

1 Introduction

In [1], [2], the multi-valued solutions of the eikonal equation were studied. Later, in [3], [4] Jin et al. provided a level set method for the computation of multi-valued geometric solutions to general quasilinear partial differential equations and multi-valued physical observables to the semiclassical limit of the Schrödinger equations. In [5], Caffarelli and Li investigated the multi-valued solutions of the Monge-Ampère equation where they first introduced the geometric situation of the multi-valued solutions and obtained the existence, regularity and the asymptotic behavior at infinity of the multi-valued viscosity solutions. In [6] Ferrer et al. used complex variable methods to study the multi-valued solutions for the Dirichlet problems of Monge-Ampère equations on exterior planar domains. Recently, Bao and Dai discussed the multi-valued solutions of Hessian equations, see [7], [8]. Motivated by the above works, in this paper we study the viscosity multi-valued solutions of the Hessian equation
F ( λ ( D 2 u ) ) = σ > 0 ,
(1.1)
where σ is a constant and λ ( D 2 u ) = ( λ 1 , λ 2 , , λ n ) are eigenvalues of the Hessian matrix D 2 u . F is assumed to be defined in the symmetric open convex cone Γ , with vertex at the origin, containing
Γ + = { λ R n : each component of  λ , λ i > 0 , i = 1 , 2 , , n } ,
satisfies the fundamental structure conditions
F i ( λ ) = F λ i > 0 in  Γ , 1 i n ,
(1.2)
and F is a continuous concave function. In addition, F will be assumed to satisfy some more technical assumptions such as
F > 0 in  Γ , F = 0 on  Γ ,
(1.3)
and for any r 1 , R > 0 ,
F ( R ( 1 r n 1 , r , , r ) ) F ( R ( 1 , 1 , , 1 ) ) .
(1.4)
For every C > 0 and every compact set K in Γ , there is Λ = Λ ( C , K ) such that
F ( Λ λ ) C for all  λ K .
(1.5)
There exists a number Λ sufficiently large such that at every point x Ω , if x 1 , , x n 1 represent the principal curvatures of Ω , then
( x 1 , , x n 1 , Λ ) Γ .
(1.6)

Inequality (1.4) is satisfied by each k th root of an elementary symmetric function ( 1 k n ) and the ( k l ) th root of each quotient of the k th elementary symmetric function and the l th elementary symmetric function ( 1 l < k n ).

2 Preliminaries

The geometric situation of the multi-valued function is given in [5]. Let n 2 , D R n be a bounded domain with smooth boundary D , and let Σ D be homeomorphic in R n to an n 1 dimensional closed disc. Σ is homeomorphic to an n 2 dimensional sphere for n 3 .

Let Z be the set of integers and M = ( D Σ ) × Z denote a covering of D Σ with the following standard parametrization: fixing x D Σ and connecting x by a smooth curve in D Σ to a point x in D Σ . If the curve goes through Σ m 0 times in the positive direction (fixing such a direction), then we arrive at ( x , m ) in M . If the curve goes through Σ m 0 times in the negative direction, then we arrive at ( x , m ) in M .

For k = 2 , 3 ,  , we introduce an equivalence relation ‘ k ’ on M as follows: ( x , m ) and ( y , j ) in M are ‘ k ’ equivalent if x = y and m j is an integer multiple of k . We let M k = M / k denote the k -sheet cover of D Σ , and let M k = m = 1 k ( D × { m } ) .

We define a distance in M k as follows: for any ( x , m ) , ( y , j ) M k , let l ( ( x , m ) , ( y , j ) ) denote a smooth curve in M k which connects ( x , m ) and ( y , j ) , and let | l ( ( x , m ) , ( y , j ) ) | denote its length. Define
d ( ( x , m ) , ( y , j ) ) = inf l | l ( ( x , m ) , ( y , j ) ) | ,

where the infimum is taken over all smooth curves connecting ( x , m ) and ( y , j ) . Then d ( ( x , m ) , ( y , j ) ) is a distance.

Definition 2.1

We say that a function u is continuous at ( x , m ) in M k if
lim d ( ( x , m ) , ( y , j ) ) 0 u ( y , j ) = u ( x , m ) ,

and u C 0 ( M k ) if for any ( x , m ) , u is continuous at ( x , m ) .

Similarly, we can define u C α ( M k ) , C 0 , 1 ( M k ) and C 2 ( M k ) .

Definition 2.2

A function u C 2 ( M k ) is called admissible if λ Γ ¯ , where λ = λ ( D 2 u ( x , m ) ) = ( λ 1 , λ 2 , , λ n ) are the eigenvalues of the Hessian matrix D 2 u ( x , m ) .

Definition 2.3

A function u C 0 ( M k ) is called a viscosity subsolution (resp. supersolution) to (1.1) if for any ( y , m ) M k and ξ C 2 ( M k ) satisfying
u ( x , m ) ( resp. ) ξ ( x , m ) , ( x , m ) M k and u ( y , m ) = ξ ( y , m ) ,
we have
F ( λ ( D 2 ξ ( y , m ) ) ) ( resp. ) σ .

Definition 2.4

A function u C 0 ( M k ) is called a viscosity solution to (1.1) if it is both a viscosity subsolution and a viscosity supersolution to (1.1).

Definition 2.5

A function u C 0 ( M k ) is called admissible if for any ( y , m ) M k and any function ξ C 2 ( M k ) satisfying u ( x , m ) ( ) ξ ( x , m ) , x M k , u ( y , m ) = ξ ( y , m ) , we have λ ( D 2 ξ ( y , m ) ) F .

Remark

It is obvious that if u is a viscosity subsolution, then u is admissible.

Lemma 2.1

Let Ω be a bounded strictly convex domain in R n , Ω C 2 , φ C 2 ( Ω ¯ ) . Then there exists a constant C only dependent on n , φ and Ω such that for any ξ Ω , there exists x ¯ ( ξ ) R n 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 n and R ¯ is a constant satisfying F ( R ¯ , R ¯ , , R ¯ ) = σ .

This is a modification of Lemma 5.1 in [5].

Lemma 2.2

Let Ω be a domain in R n and f C 0 ( R n ) be nonnegative. Assume that the admissible functions v C 0 ( Ω ¯ ) , u C 0 ( R n ) satisfy, respectively,
F ( λ ( D 2 v ) ) f ( x ) , x Ω , F ( λ ( D 2 u ) ) f ( x ) , x R n .
Moreover,
u v , x Ω ¯ , u = v , x Ω .
Set
w ( x ) = { v ( x ) , x Ω , u ( x ) , x R n Ω .
Then w C 0 ( R n ) is an admissible function and satisfies in the viscosity sense
F ( λ ( D 2 w ( x ) ) ) f ( x ) , x R n .

Lemma 2.3

Let B be a ball in R n and let f C 0 , α ( B ¯ ) be positive. Suppose that u ̲ C 0 ( B ¯ ) satisfies in the viscosity sense
F ( λ ( D 2 u ) ) f ( x ) , x B .
Then the Dirichlet problem
F ( λ ( D 2 u ) ) = f ( x ) , x B , u = u ̲ ( x ) , x B

admits a unique admissible viscosity solution u C 0 ( B ¯ ) .

We refer to [9] for the proof of Lemmas 2.2 and 2.3.

3 Existence of viscosity multi-valued solutions with asymptotic behavior

In this section, we establish the existence of viscosity multi-valued solutions with prescribed asymptotic behavior at infinity of (1.1). Let Ω be a bounded strictly convex domain with smooth boundary Ω . Let Σ , diffeomorphic to an ( n 1 ) -disc, be the intersection of Ω any hyperplane in R n . Let M = ( R n Σ ) × Z , M k = M / k be covering spaces of R n Σ as in Section 2. Σ divides Ω into two open parts, denoted as Ω + and Ω . Fixing x Ω , we use the convention that going through Σ from Ω to Ω + denotes the positive direction through Σ . Our main result is the following theorem.

Theorem 3.1

Let k 3 . Then, for any C m R , there exists an admissible viscosity solution u C 0 ( M k ) of
F ( λ ( D 2 u ) ) = σ , ( x , m ) M k
(3.1)
satisfying
lim sup | x | | x | n 2 | u ( x , m ) ( R ¯ 2 | x | 2 + C m ) | < + ,
(3.2)

where R ¯ is a constant satisfying F ( R ¯ , R ¯ , , R ¯ ) = σ .

When
F ( λ ( D 2 u ) ) = σ k ( λ ( D 2 u ) ) , Γ = Γ k = { λ R n : σ j > 0 , j = 1 , 2 , , k } ,
where the k th elementary symmetric function
σ k ( λ ) = i 1 < < i k λ i 1 λ i k

for λ = ( λ 1 , , λ n ) , in [8] Dai obtained the following result.

Theorem 3.2

Let k 3 . Then, for any C m R , there exists a k -convex viscosity solution u C 0 ( M k ) of
σ k ( λ ( D 2 u ) ) = 1 , ( x , m ) M k
satisfying
lim sup | x | ( | x | k 2 | u ( x , m ) ( C 2 | x | 2 + C m ) | ) < ,

where C = ( 1 C n k ) 1 k .

Proof of Theorem 3.1

We divide the proof of Theorem 3.1 into two steps.

Step 1. By [10], there is an admissible solution Φ C ( Ω ¯ ) of the Dirichlet problem:
F ( λ ( D 2 Φ ) ) = C 0 > σ , x Ω , Φ = 0 , x Ω .
By the comparison principles in [11], Φ 0 in Ω . Further, by Lemma 2.1, for each ξ Ω , there exists x ¯ ( ξ ) R n such that
W ξ ( x ) < Φ ( x ) , x Ω ¯ { ξ } ,
where
W ξ ( x ) = R ¯ 2 ( | x x ¯ ( ξ ) | 2 | ξ x ¯ ( ξ ) | 2 ) , ξ R n ,
and sup ξ Ω | x ¯ ( ξ ) | < . Therefore
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 n .
Denote
W ( x ) = sup ξ Ω W ξ ( x ) .
Then
W ( x ) Φ ( x ) , x Ω ,
and by [12]
F ( λ ( D 2 W ) ) σ , x R n .
Define
V ( x ) = { Φ ( x ) , x Ω , W ( x ) , x R n Ω .
Then V C 0 ( R n ) is an admissible viscosity solution of
F ( λ ( D 2 V ) ) σ , x R n .

Fix some R 1 > 0 such that Ω ¯ B R 1 ( 0 ) , where B R 1 ( 0 ) is the ball centered at the origin with radius R 1 .

Let R 2 = 2 R 1 R ¯ 1 2 . For a > 1 , defuse
W a ( x ) = inf B R 1 V + 2 R 2 | R ¯ 1 2 x | ( s n + a ) 1 n d s , x R n .
Then
D i j W a = ( | y | n + a ) 1 n 1 [ ( | y | n 1 + a | y | ) R ¯ δ i j a R ¯ 2 x i x j | y | 3 ] , | x | > 0 ,
where y = R ¯ 1 2 x . By rotating the coordinates, we may set x = ( r , 0 , , 0 ) . Therefore
D 2 W a = ( R n + a ) 1 n 1 R ¯ diag ( R n 1 , R n 1 + a R , , R n 1 + a R ) ,
where R = | y | . Consequently, λ ( D 2 W a ) Γ for | x | > 0 and by (1.4)
F ( λ ( D 2 W a ) ) F ( R ¯ , R ¯ , , R ¯ ) = σ , | x | > 0 .
Moreover,
W a ( x ) V ( x ) , | x | R 1 .
(3.3)
Fix some R 3 > 3 R 2 satisfying
R 3 R ¯ 1 2 > 3 R 2 .
We choose a 1 > 1 such that for a a 1 ,
W a ( x ) > inf B R 1 V + 2 R 2 3 R 2 ( s n + a ) 1 n d s V ( x ) , | x | = R 3 .
Then by (3.3) R 3 R 1 . According to the definition of W a ,
W a ( x ) = inf B R 1 V + 2 R 2 | R ¯ 1 2 x | s ( ( 1 + a s n ) 1 n 1 ) d s + 2 R 2 | R ¯ 1 2 x | s d s = R ¯ 2 | x | 2 + C m + inf B R 1 V + 2 R 2 + s ( ( 1 + a s n ) 1 n 1 ) d s C m 2 R 2 2 | R ¯ 1 2 x | + s ( ( 1 + a s n ) 1 n 1 ) d s , x R n .
Let
μ ( m , a ) = inf B R 1 V + 2 R 2 + s ( ( 1 + a s n ) 1 n 1 ) d s C m 2 R 2 2 .
Then μ ( m , a ) is continuous and monotonic increasing for a and when a , μ ( m , a ) , 1 m k . Moreover,
W a ( x ) = R ¯ 2 | x | 2 + C m + μ ( m , a ) O ( | x | 2 n ) , when  | x | .
(3.4)
Define, for a a 1 and 1 m k ,
u ̲ m , a ( x ) = { max { V ( x ) , W a ( x ) } μ ( m , a ) , | x | R 3 , W a μ ( m , a ) , | x | R 3 .
Then by (3.4), for 1 m k ,
u ̲ m , a ( x ) = R ¯ 2 | x | 2 + C m O ( | x | 2 n ) , when  | x | ,
and by the definition of V ,
u ̲ m , a ( x ) = μ ( m , a ) , x Σ .
Choose a 2 a 1 large enough such that when a a 2 ,
V ( x ) μ ( m , a ) = V ( x ) inf B R 1 V 2 R 2 + s ( ( 1 + a s n ) 1 n 1 ) d s + C m + 2 R 2 2 C m R ¯ 2 | x | 2 + C m , | x | R 3 .
Therefore
u ̲ m , a ( x ) R ¯ 2 | x | 2 + C m , a a 2 , x R n .
By Lemma 2.2, u ̲ m , a C 0 ( R n ) is admissible and satisfies in the viscosity sense
F ( λ ( D 2 u ̲ m , a ) ) σ , x R n .
It is easy to see that there exists a continuous function a ( m ) ( a ) such that lim a a ( m ) ( a ) = and μ ( m , a ( m ) ( a ) ) = μ ( 1 , a ) for 2 m k . So there exists a 3 a 2 such that a ( m ) ( a ) > a 2 whenever a a 3 and 2 m k . Let a ( 1 ) ( a ) = a and define
u ̲ a ( x , m ) = u ̲ m , a ( m ) ( a ) ( x ) , ( x , m ) M k .
Then, by the definition of u ̲ m , a , when a a 3 , u ̲ a C 0 ( M k ) is a locally admissible function satisfying
u ̲ a ( x , m ) = R ¯ 2 | x | 2 + C m O ( | x | 2 n ) , when  | x | , u ̲ a ( x , m ) R ¯ 2 | x | 2 + C m , x R n , 1 m k , lim x x ¯ u ̲ a ( x , m ) = μ ( 1 , a ) , x ¯ Σ , 1 m k ,
and in the viscosity sense
F ( λ ( D 2 u ̲ a ) ) σ , ( x , m ) M k .

Step 2. We define the solution of (3.1) by the Perron method.

For a a 3 , let S a denote the set of admissible functions V C 0 ( M k ) which can be extended to Σ and satisfies
F ( λ ( D 2 V ) ) σ , ( x , m ) M k , lim x x ¯ V ( x , m ) μ ( 1 , a ) , x ¯ Γ , V ( x , m ) R ¯ 2 | x | 2 + C m , x R n , 1 m k .
It is obvious that u ̲ a S a . Hence S a . Define
u a ( x , m ) = sup { V ( x , m ) : V S a } , ( x , m ) M k .
Next we prove that u a is a viscosity solution of (3.1). From the definition of u a , it is a viscosity subsolution of (3.1) and satisfies
u a ( x , m ) R ¯ 2 | x | 2 + C m , x R n .

So we need only to prove that u a is a viscosity supersolution of (3.1) satisfying (3.2).

For any x 0 R n Σ , fix ε > 0 such that B ¯ = B ε ( x 0 ) ¯ R n Σ . Then the lifting of B into M k is the k disjoint balls denoted as { B ( i ) } i = 1 k . For any ( x , m ) B ( i ) , by Lemma 2.3, there exists an admissible viscosity solution u ˜ C 0 ( B ( i ) ¯ ) to the Dirichlet problem
F ( λ ( D 2 u ˜ ) ) = σ , ( x , m ) B ( i ) , u ˜ = u a , ( x , m ) B ( i ) .
By the comparison principle in [11],
u a u ˜ , ( x , m ) B ( i ) .
(3.5)
Define
ψ ( x , m ) = { u ˜ ( x , m ) , ( x , m ) B ( i ) , u a ( x , m ) , ( x , m ) M k { B ( i ) } i = 1 k .
By Lemma 2.2,
F ( λ ( D 2 ψ ( x , m ) ) ) σ , x R n .
As
F ( λ ( D 2 u ˜ ) ) = σ = F ( λ ( D 2 g ) ) , ( x , m ) B ( i ) , u ˜ = u a g , ( x , m ) B ( i ) ,
where g ( x , m ) = R ¯ 2 | x | 2 + C m , we have
u ˜ g , ( x , m ) B ( i ) ¯

by the comparison principle in [11]. Therefore ψ S a .

By the definition of u a , u a ψ in M k . Consequently, u ˜ u a in B ( i ) and further u ˜ = u a , ( x , m ) B ( i ) in view of (3.5). Since x 0 is arbitrary, we conclude that u a is an admissible viscosity solution of (3.1).

By the definition of u a ,
u ̲ a u a g , ( x , m ) M k ,

so u a satisfies (3.2) and we complete the proof of Theorem 3.1. □

Declarations

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11371110).

Authors’ Affiliations

(1)
Department of Mathematics, Harbin Institute of Technology

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© Meng and Fu; licensee Springer 2014

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