- Open Access
Existence of viscosity multi-valued solutions with asymptotic behavior for Hessian equations
© Meng and Fu; licensee Springer 2014
- Received: 5 March 2014
- Accepted: 25 June 2014
- Published: 24 September 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.
- Hessian equation
- multi-valued solution
- asymptotic behavior
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 .
where the infimum is taken over all smooth curves connecting and . Then is a distance.
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 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.
whereforandis a constant satisfying.
This is a modification of Lemma 5.1 in .
admits a unique admissible viscosity solution.
We refer to  for the proof of Lemmas 2.2 and 2.3.
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.
whereis a constant satisfying.
for , in  Dai obtained the following result.
Proof of Theorem 3.1
We divide the proof of Theorem 3.1 into two steps.
Fix some such that , where is the ball centered at the origin with radius .
Step 2. We define the solution of (3.1) by the Perron method.
So we need only to prove that is a viscosity supersolution of (3.1) satisfying (3.2).
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).
so satisfies (3.2) and we complete the proof of Theorem 3.1. □
This work was supported by the National Natural Science Foundation of China (Grant No. 11371110).
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