Editing Elliptic partial differential equation (section)

==Definition==
Elliptic differential equations appear in many different contexts and levels of generality. 

First consider a second-order linear PDE for an unknown function of two variables <math>u = u(x,y)</math>, written in the form
<math display="block">Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu +G= 0,</math>
where {{math|''A''}}, {{math|''B''}}, {{math|''C''}}, {{math|''D''}}, {{math|''E''}}, {{math|''F''}}, and {{math|''G''}} are functions of <math>(x,y)</math>, using [[Notation_for_differentiation#Partial_derivatives|subscript notation]] for the partial derivatives. The PDE is called '''elliptic''' if
<math display="block">B^2-AC<0,</math>
by analogy to the equation for a [[ellipse#General ellipse|planar ellipse]]. Equations with <math>B^2 - AC = 0</math> are termed [[Parabolic partial differential equation|parabolic]] while those with <math>B^2 - AC > 0</math> are [[Hyperbolic partial differential equation|hyperbolic]].

For a general linear second-order PDE, the unknown {{mvar|u}} can be a function of any number of independent variables, <math>u=u(x_1,\ldots,x_n)</math>, satisfying an equation of the form
<math display="block">\sum_{i=1}^n\sum_{j=1}^n a_{ij}(x_1,\ldots,x_n) u_{x_i x_j} +\sum_{i=1}^n b_i(x_1,\ldots,x_n) u_{x_i} +c(x_1,\ldots,x_n)u=f(x_1,\ldots,x_n).</math>
where <math>a_{ij}, b_i, c, f</math> are functions defined on the domain subject to the symmetry <math>a_{ij}=a_{ji}</math>. This equation is called '''elliptic''' if, viewing <math>a=(a_{ij})</math> as a function of <math>(x_1,\ldots,x_n)</math> valued in the space of <math>n\times n</math> [[symmetric matrix|symmetric matrices]], all [[eigenvalues]] are greater than some positive constant: that is, there is a positive number {{mvar|&theta;}} such that
<math display="block">\sum_{i=1}^n\sum_{j=1}^n a_{ij}(x_1,\ldots,x_n)\xi_i\xi_j\geq \theta(\xi_1^2+\cdots+\xi_n^2)</math>
for every point <math>(x_1,\ldots,x_n)</math> in the domain and all real numbers {{math|&xi;<sub>1</sub>, ..., &xi;<sub>''n''</sub>}}.{{sfnm|1a1=Evans|1y=2010|1loc=Chapter 6}}{{sfn|Zauderer|2006|loc=chpt. 3.3 Classification of equations in general}}

The simplest example of a second-order linear elliptic PDE is the [[Laplace equation]], in which the coefficients are the constant functions <math>a_{ij}=0</math> for <math>i\neq j</math>, <math>a_{ii}=1</math>, and <math>b_i=c=f=0</math>. The [[Poisson equation]] is a slightly more general second-order linear elliptic PDE, in which {{mvar|f}} is not required to vanish. For both of these equations, the ellipticity constant {{mvar|&theta;}} can be taken to be {{math|1}}.

The terminology is not used consistently throughout the literature: what is called "elliptic" by some authors is called "strictly elliptic" or "uniformly elliptic''"'' by others.<ref>Compare {{harvtxt|Evans|2010|p=311}} and {{harvtxt|Gilbarg|Trudinger|2001|pp=31,441}}.</ref>
===Nonlinear and higher-order equations===
{{Broader|Elliptic operator}}

Ellipticity can also be formulated for much more general classes of equations. For the most general second-order PDE, which is of the form
:<math>F(D^2u,Du,u,x_1,\ldots,x_n)=0</math>
for some given function {{mvar|F}}, '''ellipticity''' is defined by [[linearization|linearizing]] the equation and applying the above linear definition. Since linearization is done at a particular function {{mvar|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, the simplest [[Monge–Ampère equation]] involves the [[determinant]] of the [[hessian matrix]] of the unknown function:
:<math> \det D^2u = f.</math>
As follows from [[Jacobi's formula]] for the derivative of a determinant, this equation is elliptic if {{mvar|f}} is a positive function and solutions satisfy the constraint of being [[uniformly convex]].{{sfnm|1a1=Gilbarg|1a2=Trudinger|1y=2001|1loc=Chapter 17}}

There are also higher-order elliptic PDE, the simplest example being the fourth-order [[biharmonic equation]].{{sfnm|1a1=John|1y=1982|1loc=Chapter 6|2a1=Ladyzhenskaya|2y=1985|2loc=Section V.1|3a1=Renardy|3a2=Rogers|3y=2004|3loc=Section 9.1}} Even more generally, there is an important class of ''elliptic systems'' which consist of coupled partial differential equations for multiple unknown functions.{{sfnm|1a1=Agmon|1y=2010|2a1=Morrey|2y=1966}} For example, the [[Cauchy–Riemann equation]]s from [[complex analysis]] can be viewed as a first-order elliptic system for a pair of two-variable real functions.{{sfnm|1a1=Courant|1a2=Hilbert|1y=1962|1p=176}}

Moreover, the class of elliptic PDE (of any order, including systems) is subject to various notions of [[weak solution]]s, i.e., reformulating the equations in a way that allows for solutions with various irregularities (e.g. [[Derivative|non-differentiability]], [[Singularity_(mathematics)|singularities]] or [[Classification_of_discontinuities|discontinuities]]), so as to model non-smooth physical phenomena.{{sfnm|1a1=Crandall|1a2=Ishii|1a3=Lions|1y=1992|2a1=Evans|2y=2010|2loc=Chapter 6|3a1=Gilbarg|3a2=Trudinger|3y=2001|3loc=Chapters 8 and 9|4a1=Ladyzhenskaya|4y=1985|4loc=Sections II.2 and V.1|5a1=Renardy|5a2=Rogers|5y=2004|5loc=Chapter 9}} Such solutions are also important in [[variational calculus]], where the [[direct method in the calculus of variations|direct method]] often produces weak solutions for elliptic systems of [[Euler_equations_(fluid_dynamics)#Discontinuities|Euler equation]]s.{{sfnm|1a1=Giaquinta|1y=1983|2a1=Morrey|2y=1966|2pp=8,480}}