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.
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.
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.
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).
- Gosse L, Jin S, Li X: Two moment systems for computing multiphase semiclassical limits of the Schrödinger equation. Math. Models Methods Appl. Sci. 2003, 13: 1689-1723. 10.1142/S0218202503003082MathSciNetView ArticleGoogle Scholar
- Izumiya S, Kossioris GT, Makrakis GN: Multivalued solutions to the eikonal equation in stratified media. Q. Appl. Math. 2001, 59: 365-390.MathSciNetGoogle Scholar
- Jin S, Osher S: A level set method for the computation of multivalued solutions to quasi-linear hyperbolic PDEs and Hamilton-Jacobi equations. Commun. Math. Sci. 2003, 1: 575-591. 10.4310/CMS.2003.v1.n3.a9MathSciNetView ArticleGoogle Scholar
- Jin S, Liu H, Osher S, Tsai Y: Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation. J. Comput. Phys. 2005, 205: 222-241. 10.1016/j.jcp.2004.11.008MathSciNetView ArticleGoogle Scholar
- Caffarelli L, Li YY: Some multi-valued solutions to Monge-Ampère equations. Commun. Anal. Geom. 2006, 14: 411-441. 10.4310/CAG.2006.v14.n3.a1MathSciNetView ArticleGoogle Scholar
- Ferrer L, Martínez A, Milán F: An extension of a theorem by K. Jörgens and a maximum principle at infinity for parabolic affine spheres. Math. Z. 1999, 230: 471-486. 10.1007/PL00004700MathSciNetView ArticleGoogle Scholar
- Dai LM, Bao JG: Multi-valued solutions to Hessian equations. Nonlinear Differ. Equ. Appl. 2011, 18: 447-457. 10.1007/s00030-011-0103-8MathSciNetView ArticleGoogle Scholar
- Dai LM: Existence of multi-valued solutions with asymptotic behavior of Hessian equations. Nonlinear Anal. 2011, 74: 3261-3268. 10.1016/j.na.2011.02.004MathSciNetView ArticleGoogle Scholar
- Tian BP, Fu YQ: Existence of viscosity solutions for Hessian equations in exterior domains. Front. Math. China 2014, 9: 201-211. 10.1007/s11464-013-0340-8MathSciNetView ArticleGoogle Scholar
- Caffarelli L, Nirenberg L, Spruck J: The Dirichlet problem for nonlinear second-order elliptic equations III: functions of eigenvalues of the Hessians. Acta Math. 1985, 15: 261-301. 10.1007/BF02392544MathSciNetView ArticleGoogle Scholar
- Trudinger NS: The Dirichlet problem for the prescribed curvature equations. Arch. Ration. Mech. Anal. 1990, 111: 153-179. 10.1007/BF00375406MathSciNetView ArticleGoogle Scholar
- Ishii H, Lions PL: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equ. 1990, 83: 26-78. 10.1016/0022-0396(90)90068-ZMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.