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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 Î“,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>0in Î“,F=0on âˆ‚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(Λλ)≥Cfor all Î»âˆˆK.
(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 kth root of an elementary symmetric function (1≤k≤n) and the (k−l)th root of each quotient of the kth elementary symmetric function and the lth 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 Σm≥0 times in the positive direction (fixing such a direction), then we arrive at (x,m) in M. If the curve goes through Σm≥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 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 anyξ∈∂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 )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),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 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 (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

Letk≥3. 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

Letk≥3. 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], 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 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 ds≥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 ) ds− 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 Ëœ ∈ 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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Meng, X., Fu, Y. Existence of viscosity multi-valued solutions with asymptotic behavior for Hessian equations. Bound Value Probl 2014, 165 (2014). https://doi.org/10.1186/s13661-014-0165-8

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