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

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.

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 >0in Γ,1in,
(1.2)

and F is a continuous concave function. In addition, F will be assumed to satisfy some more technical assumptions such as

F>0in Γ,F=0on Γ,
(1.3)

and for any r1, 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(Λλ)Cfor all λK.
(1.5)

There exists a number Λ sufficiently large such that at every point xΩ, 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 kth root of an elementary symmetric function (1kn) and the (kl)th root of each quotient of the kth elementary symmetric function and the lth elementary symmetric function (1l<kn).

2 Preliminaries

The geometric situation of the multi-valued function is given in [5]. Let n2, D R n be a bounded domain with smooth boundary D, and let ΣD be homeomorphic in R n to an n1 dimensional closed disc. Σ is homeomorphic to an n2 dimensional sphere for n3.

Let Z be the set of integers and M=(DΣ)×Z denote a covering of DΣ with the following standard parametrization: fixing x DΣ and connecting x by a smooth curve in DΣ to a point x in DΣ. If the curve goes through Σm0 times in the positive direction (fixing such a direction), then we arrive at (x,m) in M. If the curve goes through Σm0 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 mj is an integer multiple of k. We let M k =M/k denote the k-sheet cover of DΣ, 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 andu(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

LetΩbe a bounded strictly convex domain in R n , Ω C 2 , φ C 2 ( Ω ¯ ). Then there exists a constantConly dependent onn, φandΩsuch that for anyξΩ, 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 )forx R n and R ¯ is a constant satisfyingF( R ¯ , R ¯ ,, R ¯ )=σ.

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

Lemma 2.2

LetΩbe a domain in R n andf C 0 ( R n )be nonnegative. Assume that the admissible functionsv 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

LetBbe a ball in R n and letf C 0 , α ( B ¯ )be positive. Suppose that u ̲ C 0 ( B ¯ )satisfies in the viscosity sense

F ( λ ( D 2 u ) ) f(x),xB.

Then the Dirichlet problem

F ( λ ( D 2 u ) ) = f ( x ) , x B , u = u ̲ ( x ) , x B

admits a unique admissible viscosity solutionu 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 (n1)-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

Letk3. Then, for any C m R, there exists an admissible viscosity solutionu 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 satisfyingF( 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 kth 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

Letk3. Then, for any C m R, there exists ak-convex viscosity solutionu 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], Φ0 in Ω. Further, by Lemma 2.1, for each ξΩ, 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),xΩ,

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 ds,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 dsV(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 ) ds C m 2 R 2 2 .

Then μ(m,a) is continuous and monotonic increasing for a and when a, μ(m,a), 1mk. 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 1mk,

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 1mk,

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),xΣ.

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 2mk. So there exists a 3 a 2 such that a ( m ) (a)> a 2 whenever a a 3 and 2mk. 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. □

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Acknowledgements

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

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Correspondence to Yongqiang Fu.

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All authors contributed to each part of this work equally and read and approved the final version of the manuscript.

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Keywords

  • Hessian equation
  • multi-valued solution
  • asymptotic behavior