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.{{citation | first1 = Andrea | last1 = Colesanti | title = On entire solutions of the Hessian equations S_k(D²u) = 1 | journal = Quaderno del Dipartimento di Matematica "U. Dini", Universitá degli Studi di Firenze | year = 2004 | url = http://web.math.unifi.it/users/colesant/ricerca/slzint.pdf }}. It can be written as $\{\backslash cal\; S\}\_k[u]=f$, where $1\backslash leqslant\; k\; \backslash leqslant\; n$, $\{\backslash cal\; S\}\_k[u]=\backslash sigma\_k(\backslash lambda(\{\backslash cal\; D\}^2u))$, and $\backslash lambda(\{\backslash cal\; D\}^2u)=(\backslash lambda\_1,\backslash cdots,\backslash lambda\_n)$, are the eigenvalues of the Hessian matrix $\{\backslash cal\; D\}^2u=[\backslash partial\_i\; \backslash partial\_ju]\_\{1\backslash leq\; i,j\; \backslash leq\; n\}$ and

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{{citation | last1 = Wang | first1 = Xu-Jia | chapter = The k-Hessian Equation | chapter-url = http://maths-people.anu.edu.au/~wang/publications/k-Hessian.pdf | editor1-first = Sun-Yung Alice | editor1-last = Chang | editor2-first = Antonio | editor2-last = Ambrosetti | editor3-first = Andrea | editor3-last = Malchiodi | title = Geometric Analysis and PDEs | series = Lecture Notes in Mathematics | volume = 1977 | year = 2009 | publisher = Springer-Verlag | isbn = 978-3-642-01673-8 }}. and Poisson's equation (the Laplacian being the trace of the Hessian matrix). The 2−hessian operator also appears in conformal mapping problems. In fact, the 2−hessian equation is unfamiliar outside Riemannian geometry and elliptic regularity theory, that is closely related to the scalar curvature operator, which provides an intrinsic curvature for a three-dimensional manifold.

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