The Hessian equation on nonsmooth k-convex domains or in the presence of subsolutions

2026/07/23 by J. Lukas Gehring
#math.AP

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Abstract

It is proved for k>n/2 that the Dirichlet problem of the k-Hessian measure on a (nonsmooth and nonuniformly) k-convex (also known as k-hyperconvex) domain for any finite Borel measure as right-hand side and boundary data taken from a k-convex and continuous function on the closure of the domain has a unique solution. Merely continuous boundary data are sufficient for strictly k-convex domains (i.e., if there exist strong barriers).

Citations

• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian

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• The Continuous Subsolution Problem for Complex Hessian Equations