Editing Elliptic partial differential equation (section)

==Canonical form==
Consider a second-order elliptic partial differential equation
:<math>A(x,y)u_{xx}+2B(x,y)u_{xy}+C(x,y)u_{yy}+f(u_x,u_y,u,x,y)=0</math>
for a two-variable function <math>u=u(x,y)</math>. This equation is linear in the "leading" highest-order terms, but allows nonlinear expressions involving the function values and their first derivatives; this is sometimes called a [[Partial differential equation#Linear and nonlinear equations|quasilinear equation]].

A ''canonical form'' asks for a transformation <math>(w,z) = (w(x,y), z(x,y))</math> of the <math>(x,y)</math> domain so that, when {{mvar|u}} is viewed as a function of {{mvar|w}} and {{mvar|z}}, the above equation takes the form
:<math>u_{ww}+u_{zz}+F(u_w,u_z,u,w,z)=0</math>
for some new function {{mvar|F}}. The existence of such a transformation can be established ''locally'' if {{mvar|A}}, {{mvar|B}}, and {{mvar|C}} are [[real-analytic function]]s and, with more elaborate work, even if they are only [[continuously differentiable]]. Locality means that the necessary coordinate transformations may fail to be defined on the entire domain of {{mvar|u}}, only in some small region surrounding any particular point of the domain.{{sfnm|1a1=Courant|1a2=Hilbert|1y=1962}}

Formally establishing the existence of such transformations uses the existence of solutions to the [[Beltrami equation]]. From the perspective of [[differential geometry]], the existence of a canonical form is equivalent to the existence of [[isothermal coordinates]] for the associated [[Riemannian metric]]
:<math>A(x,y) dx^2+2B(x,y)\,dx\,dy+C(x,y)dy^2</math>
on the domain. (The ellipticity condition for the PDE, namely the positivity of {{math|''AC'' – ''B''<sup>2</sup>}}, is what ensures that either this tensor or its negation is indeed a Riemannian metric.) 

For second-order quasilinear elliptic partial differential equations in ''more'' than two variables, a canonical form does ''not'' usually exist. This corresponds to the fact that isothermal coordinates do not exist for general Riemannian metrics in higher dimensions, only for very particular ones.{{sfnm|1a1=Spivak|1y=1979}}

===Characteristics and regularity===
For the general second-order linear PDE, [[Partial_differential_equation#Systems_of_first-order_equations_and_characteristic_surfaces|characteristics]] are defined as the [[Pseudo-Riemannian manifold|null direction]]s for the associated tensor{{sfn|Hörmander|1990|p=152}}
:<math>\sum_{i=1}^n \sum_{j=1}^n a_{i,j}(x_1,\ldots,x_n)\,dx^i\,dx^j,</math>
called the [[principal symbol]]. Using the technology of the [[wave front set]], characteristics are significant in understanding how irregular points of {{mvar|f}} propagate to the solution {{mvar|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.{{sfnm|1a1=Hörmander|1y=1990|1p=256}} 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.{{sfnm|1a1=Hörmander|1y=1990|1loc=Theorem 8.3.1}}

In the case of a linear ''elliptic'' operator {{mvar|P}} with smooth coefficients, the principal symbol is a [[Riemannian metric]] and there are no real characteristic directions. According to the previous paragraph, it follows that the wave front set of a solution {{mvar|u}} coincides exactly with that of {{math|''Pu'' {{=}} ''f''}}. This sets up a basic ''regularity theorem'', which says that if {{math|''f''}} is smooth (so that its wave front set is empty) then the solution {{mvar|u}} is smooth as well. More generally, the points where {{mvar|u}} fails to be smooth coincide with the points where {{math|''f''}} is not smooth.{{sfnm|1a1=Hörmander|1y=1990|1loc=Corollary 8.3.2}} 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.

Solutions of [[elliptic PDE]]s are naturally associated with time-independent solutions of [[parabolic PDE]]s or [[hyperbolic PDE]]s. For example, a time-independent solution of the [[heat equation]] solves [[Laplace's equation]]. That is, if parabolic and hyperbolic PDEs are associated with modeling [[dynamical systems]] then the solutions of elliptic PDEs are associated with [[steady state]]s. 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.