Existence of viscosity multi-valued solutions with asymptotic behavior for Hessian equations
Boundary Value Problemsvolume 2014, Article number: 165 (2014)
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.
In , , the multi-valued solutions of the eikonal equation were studied. Later, in ,  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 , 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  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 , . Motivated by the above works, in this paper we study the viscosity multi-valued solutions of the Hessian equation
where is a constant and are eigenvalues of the Hessian matrix . is assumed to be defined in the symmetric open convex cone , with vertex at the origin, containing
satisfies the fundamental structure conditions
and is a continuous concave function. In addition, will be assumed to satisfy some more technical assumptions such as
and for any , ,
For every and every compact set in , there is such that
There exists a number sufficiently large such that at every point , if represent the principal curvatures of , then
Inequality (1.4) is satisfied by each th root of an elementary symmetric function () and the th root of each quotient of the th elementary symmetric function and the th elementary symmetric function ().
The geometric situation of the multi-valued function is given in . Let , be a bounded domain with smooth boundary , and let be homeomorphic in to an dimensional closed disc. is homeomorphic to an dimensional sphere for .
Let be the set of integers and denote a covering of with the following standard parametrization: fixing and connecting by a smooth curve in to a point in . If the curve goes through times in the positive direction (fixing such a direction), then we arrive at in . If the curve goes through times in the negative direction, then we arrive at in .
For , we introduce an equivalence relation ‘’ on as follows: and in are ‘’ equivalent if and is an integer multiple of . We let denote the -sheet cover of , and let .
We define a distance in as follows: for any , let denote a smooth curve in which connects and , and let denote its length. Define
where the infimum is taken over all smooth curves connecting and . Then is a distance.
We say that a function is continuous at in if
and if for any , is continuous at .
Similarly, we can define , and .
A function is called admissible if , where are the eigenvalues of the Hessian matrix .
A function is called a viscosity subsolution (resp. supersolution) to (1.1) if for any and satisfying
A function is called a viscosity solution to (1.1) if it is both a viscosity subsolution and a viscosity supersolution to (1.1).
A function is called admissible if for any and any function satisfying , , , we have .
It is obvious that if is a viscosity subsolution, then is admissible.
Letbe a bounded strictly convex domain in, , . Then there exists a constantonly dependent on, andsuch that for any, there existssuch that
whereforandis a constant satisfying.
This is a modification of Lemma 5.1 in .
Letbe a domain inandbe nonnegative. Assume that the admissible functions, satisfy, respectively,
Then is an admissible function and satisfies in the viscosity sense
Letbe a ball inand letbe positive. Suppose thatsatisfies in the viscosity sense
Then the Dirichlet problem
admits a unique admissible viscosity solution.
We refer to  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 be a bounded strictly convex domain with smooth boundary . Let , diffeomorphic to an -disc, be the intersection of any hyperplane in . Let , be covering spaces of as in Section 2. divides into two open parts, denoted as and . Fixing , we use the convention that going through from to denotes the positive direction through . Our main result is the following theorem.
Let. Then, for any, there exists an admissible viscosity solutionof
whereis a constant satisfying.
where the th elementary symmetric function
for , in  Dai obtained the following result.
Let. Then, for any, there exists a-convex viscosity solutionof
Proof of Theorem 3.1
We divide the proof of Theorem 3.1 into two steps.
Step 1. By , there is an admissible solution of the Dirichlet problem:
By the comparison principles in , in . Further, by Lemma 2.1, for each , there exists such that
and . Therefore
and by 
Then is an admissible viscosity solution of
Fix some such that , where is the ball centered at the origin with radius .
Let . For , defuse
where . By rotating the coordinates, we may set . Therefore
where . Consequently, for and by (1.4)
Fix some satisfying
We choose such that for ,
Then by (3.3) . According to the definition of ,
Then is continuous and monotonic increasing for and when , , . Moreover,
Define, for and ,
Then by (3.4), for ,
and by the definition of ,
Choose large enough such that when ,
By Lemma 2.2, is admissible and satisfies in the viscosity sense
It is easy to see that there exists a continuous function such that and for . So there exists such that whenever and . Let and define
Then, by the definition of , when , is a locally admissible function satisfying
and in the viscosity sense
Step 2. We define the solution of (3.1) by the Perron method.
For , let denote the set of admissible functions which can be extended to and satisfies
It is obvious that . Hence . Define
Next we prove that is a viscosity solution of (3.1). From the definition of , it is a viscosity subsolution of (3.1) and satisfies
So we need only to prove that is a viscosity supersolution of (3.1) satisfying (3.2).
For any , fix such that . Then the lifting of into is the disjoint balls denoted as . For any , by Lemma 2.3, there exists an admissible viscosity solution to the Dirichlet problem
By the comparison principle in ,
By Lemma 2.2,
where , we have
by the comparison principle in . Therefore .
By the definition of in . Consequently, in and further , in view of (3.5). Since is arbitrary, we conclude that is an admissible viscosity solution of (3.1).
By the definition of ,
so satisfies (3.2) and we complete the proof of Theorem 3.1. □
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This work was supported by the National Natural Science Foundation of China (Grant No. 11371110).
The authors declare that they have no competing interests.
All authors contributed to each part of this work equally and read and approved the final version of the manuscript.