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] It can be written as , where , , and , are the eigenvalues of the Hessian matrix and , is a th elementary symmetric polynomial.[2][3]
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[4] 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.
^Colesanti, Andrea (2004), "On entire solutions of the Hessian equations Sk(D2u) = 1" (PDF), Quaderno del Dipartimento di Matematica "U. Dini", Universitá degli Studi di Firenze.
^Yourdkhany, Mahdieh; Nadjafikhah, Mehdi; Toomanian, Megerdich (2021-08-01). "Preliminary Group Classification and Some Exact Solutions of the 2-Hessian Equation". Bulletin of the Iranian Mathematical Society. 47 (4): 977–994. arXiv:1902.02702. doi:10.1007/s41980-020-00424-3. ISSN 1735-8515. S2CID 225550133.
^Froese, Brittany D.; Oberman, Adam M.; Salvador, Tiago (2016-05-14). "Numerical methods for the 2-Hessian elliptic partial differential equation". IMA Journal of Numerical Analysis. 37 (1): 209–236. arXiv:1502.04969. doi:10.1093/imanum/drw007. ISSN 0272-4979.
^Wang, Xu-Jia (2009), "The k-Hessian Equation" (PDF), in Chang, Sun-Yung Alice; Ambrosetti, Antonio; Malchiodi, Andrea (eds.), Geometric Analysis and PDEs, Lecture Notes in Mathematics, vol. 1977, Springer-Verlag, ISBN 978-3-642-01673-8.
In mathematics, k-Hessianequations (or Hessianequations for short) are partial differential equations (PDEs) based on the Hessian matrix. More specifically...
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