Hessian equation

In mathematics, k-Hessian equations (or Hessian equations for short) are partial differential equations (PDEs) based on the Hessian matrix. More specifically, a Hessian equation is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k ≥ 2, the k-Hessian equation is a fully nonlinear partial differential equation.^[1]

Much like differential equations often study the actions of differential operators (e.g. elliptic operators and elliptic equations), Hessian equations can be understood as simply eigenvalue equations acted upon by the Hessian differential operator. Special cases include the Monge–Ampère equation^[2] and Poisson's equation (the Laplacian being the trace of the Hessian matrix).

These equations are of interest in geometric PDEs (a subfield at the interface between both geometric analysis and PDEs) and differential geometry.

References

1. Colesanti, Andrea (2004), "On entire solutions of the Hessian equations S_k(D²u) = 1" (PDF), Quaderno del Dipartimento di Matematica "U. Dini", Universitá degli Studi di Firenze.
2. Wang, Xu-Jia (2009), "The k-Hessian Equation" (PDF), in Chang, Sun-Yung Alice; Ambrosetti, Antonio; Malchiodi, Andrea, Geometric Analysis and PDEs, Lecture Notes in Mathematics 1977, Springer-Verlag, ISBN 978-3-642-01673-8.

Further reading

• Caffarelli, L.; Nirenberg, L.; Spruck, J. (1985), "The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian" (PDF), Acta Mathematica 155 (1): 261–301, doi:10.1007/BF02392544.