They are also important in pure mathematics, where they are fundamental to various fields of research such as differential geometry and optimal transport.
Elliptic differential equations appear in many different contexts and levels of generality.
First consider a second-order linear PDE in two variables, written in the form
with this naming convention inspired by the equation for a planar ellipse.
This equation is called elliptic if, when a is viewed as a function on the domain valued in the space of n × n symmetric matrices, all of the eigenvalues are greater than some set positive number.
for any point x1, ..., xn in the domain and any real numbers ξ1, ..., ξn.
[1][2] The simplest example of a second-order linear elliptic PDE is the Laplace equation, in which ai,j is zero if i ≠ j and is one otherwise, and where bi = c = f = 0.
The Poisson equation is a slightly more general second-order linear elliptic PDE, in which f is not required to vanish.
The terminology elliptic partial differential equation is not used consistently throughout the literature.
Since linearization is done at a particular function u, this means that ellipticity of a nonlinear second-order PDE depends not only on the equation itself but also on the solutions under consideration.
For example, in the simplest kind of Monge–Ampère equation, the determinant of the hessian matrix of a function is prescribed: As follows from Jacobi's formula for the derivative of a determinant, this equation is elliptic if f is a positive function and solutions satisfy the constraint of being uniformly convex.
[4] There are also higher-order elliptic PDE, the simplest example being the fourth-order biharmonic equation.
[5] Even more generally, there is an important class of elliptic systems which consist of coupled partial differential equations for multiple 'unknown' functions.
[6] For example, the Cauchy–Riemann equations from complex analysis can be viewed as a first-order elliptic system for a pair of two-variable functions.
[7] Moreover, the class of elliptic PDE (of any order, including systems) is subject to various notions of weak solutions, i.e., reformulating the above equations in such a way that allows for solutions to have various irregularities (e.g. non-differentiability, singularities or discontinuities) while still adhering to the laws of physics.
[8] Additionally, these type of solutions are also important in variational calculus, where the direct method often produces weak solutions of elliptic systems of Euler equations.
[9] Consider a second-order elliptic partial differential equation for a two-variable function u = u(x, y).
Locality means that the necessary coordinate transformations may fail to be defined on the entire domain of u, although they can be established in some small region surrounding any particular point of the domain.
(The ellipticity condition for the PDE, namely the positivity of the function AC – B2, is what ensures that either this tensor or its negation is indeed a Riemannian metric.)
Generally, for second-order quasilinear elliptic partial differential equations for functions of more than two variables, a canonical form does not exist.
This corresponds to the fact that, although isothermal coordinates generally exist for Riemannian metrics in two dimensions, they only exist for very particular Riemannian metrics in higher dimensions.
[11] For the general second-order linear PDE, characteristics are defined as the null directions for the associated tensor[12] called the principal symbol.
Using the technology of the wave front set, characteristics are significant in understanding how irregular points of f propagate to the solution u of the PDE.
Informally, the wave front set of a function consists of the points of non-smoothness, in addition to the directions in frequency space causing the lack of smoothness.
It is a fundamental fact that the application of a linear differential operator with smooth coefficients can only have the effect of removing points from the wave front set.
[13] However, all points of the original wave front set (and possibly more) are recovered by adding back in the (real) characteristic directions of the operator.
[14] In the case of a linear elliptic operator P with smooth coefficients, the principal symbol is a Riemannian metric and there are no real characteristic directions.
[15] This regularity phenomena is in sharp contrast with, for example, hyperbolic PDE in which discontinuities can form even when all the coefficients of an equation are smooth.
Informally, this is reflective of the above regularity theorem, as steady states are generally smoothed out versions of truly dynamical solutions.
However, PDE used in modeling are often nonlinear and the above regularity theorem only applies to linear elliptic equations; moreover, the regularity theory for nonlinear elliptic equations is much more subtle, with solutions not always being smooth.