Hyperbolic Polynomials and the Dirichlet Problem

This paper presents a simple, self-contained account of G̊arding’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 G̊arding’s theory to the authors’ recent work on the Dirichlet problem for fully nonlinear partial differential equations is investigated. Each G̊arding polynomial p of degree m on Sym(Rn) (hyperbolic with respect to the identity) has an associated eigenvalue map λ : Sym(Rn) → R, defined modulo the permutation group acting on R. Consequently, each closed symmetric set E ⊂ R induces a second-order p.d.e. by requiring, for a C-function u in n-variables, that λ ( (Du)(x) ) ∈ ∂E for all x. Assume that A ≥ 0 ⇒ λ(A) ≥ 0 and that E +R+ ⊂ E. A main result is that for smooth domains Ω ⊂ R whose boundary is suitably (p, E)pseudo-convex, the Dirichlet problem has a unique continuous solution for all continuous boundary data. This applies in particular to each of the m distinct branches of the equation p ( Du ) = 0 In the authors’ recent extension of results from euclidean domains to domains in riemannian manifolds, 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 G̊arding cone is a monotonicity cone for all branches of the the equation p(Hessu) = 0 where Hessu denotes the riemannian Hessian of u. ∗Partially supported by the N.S.F.