Hyperbolic polynomials and the Dirichlet problem

TL;DR

This paper provides a clear account of Garding's hyperbolic polynomial theory, introduces new results on eigenvalue arrangements, and explores their application to solving the Dirichlet problem for nonlinear PDEs on various domains.

Contribution

It offers a self-contained presentation of hyperbolic polynomials, establishes new eigenvalue function results, and links Garding's theory to solving nonlinear PDEs on Riemannian manifolds.

Findings

Existence of a real analytic eigenvalue arrangement.

Unique solutions to the Dirichlet problem on pseudo-convex domains.

Garding cones serve as monotonicity cones for PDE branches.

Abstract

This paper presents a simple, self-contained account of Garding's theory of hyperbolic polynomials, including a recent convexity result of Bauschke-Guler-Lewis-Sendov and an inequality of Gurvits. This account also contains new results, such as the existence of a real analytic arrangement of the eigenvalue functions. In a second, independent part of the paper, the relationship of Garding's theory to the authors' recent work (arXiv:0710.3991) on the Dirichlet problem for fully nonlinear partial differential equations is investigated. Let p be a homogeneous polynomial of degree m on S^2(R^n) which is hyperbolic with respect to the all positive directions A \geq 0. Then p has an associated eigenvalue map lambda:S^2(R^n) \to R^m, defined modulo the permutation group acting on R^m. Consequently, each closed symmetric set E of R^m induces a second-order p.d.e. by requiring, for a C^2-function u in n-variables, that (D^2 u)(x) lie in the boundary of E for all x. Assume that E + (R_+)^m is contained in E. A main result is that for smooth domains in R^n whose boundary is suitably (p,E)-pseudo-convex, the Dirichlet problem has a unique continuous solution for all continuous boundary data. This applies to a vast collection of examples the most basic of which are the m distinct branches of the equation p(D^2 u) =0. In the authors' recent extension of results from euclidean domains to domains in riemannian manifolds (arXiv:0907.1981), a new global ingredient, called a monotonicity subequation, was introduced. It is shown in this paper that for every polynomial $p$ as above, the associated Garding cone is a monotonicity cone for all branches of the the equation p(Hess u) = 0 where Hess u denotes the riemannian Hessian of u.