In mathematics, ''k''-Hessian equations (or Hessian equations for short) are

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...

s (PDEs) based on the

Hessian matrix In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...

. More specifically, a Hessian equation is the ''k''-trace, or the ''k''th

elementary symmetric polynomial In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sym ...

of eigenvalues of the Hessian matrix. When ''k'' ≥ 2, the ''k''-Hessian equation is a fully nonlinear partial differential equation. It can be written as

_k f

, where

1\leqslant k \leqslant n

\sigma_k(\lambda(^2u))

, and

\lambda(^2u)=(\lambda_1,\cdots,\lambda_n)

, are the eigenvalues of the ''Hessian matrix''

^2u= partial_i \partial_ju

and

\sigma_k(\lambda)=\sum_\lambda_\cdots\lambda_

, is a

k

th elementary symmetric polynomial. Much like differential equations often study the actions of

differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...

s (e.g.

elliptic operator In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which i ...

s 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 In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function ''u'' of two variables ''x'',''y'' is of Monge–Ampère type if it is li ...

. and

Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...

(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 Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of l ...

and PDEs) and

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

References

Further reading