Inmathematics, anelliptic partial differential equation is a type ofpartial differential equation (PDE). Inmathematical modeling, elliptic PDEs are frequently used to modelsteady states, unlikeparabolic andhyperbolic PDEs, which generally model phenomena that change in time. The canonical examples of elliptic PDEs areLaplace's equation andPoisson's equation. Elliptic PDEs are also important inpure mathematics, where they are fundamental to various fields of research such asdifferential geometry andoptimal transport.

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${\displaystyle u=u(x,y)}$, written in the form$${\displaystyle A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}+D{u}_x+E{u}_y+Fu+G=0,}$$whereA,B,C,D,E,F, andG are functions of${\displaystyle (x,y)}$, usingsubscript notation for the partial derivatives. The PDE is calledelliptic ifby analogy to the equation for aplanar ellipse. Equations with${\displaystyle B^2-AC=0}$ are termedparabolic while those with${\displaystyle B^2-AC>0}$ arehyperbolic.

For a general linear second-order PDE, the unknown can be a function of any number of independent variables${\displaystyle u=u(x_1,\dots ,x_n)}$, satisfying an equation of the form$${\displaystyle \sum _{i=1}^n\sum _{j=1}^n{a}_{ij}(x_1,\dots ,x_n)u_{x_i{x}_j}+\sum _{i=1}^n{b}_i(x_1,\dots ,x_n)u_{x_i}+c(x_1,\dots ,x_n)u=f(x_1,\dots ,x_n).}$$where${\displaystyle a_{ij},b_i,c,f}$ are functions defined on the domain subject to the symmetry${\displaystyle a_{ij}=a_{ji}}$. This equation is calledelliptic if, viewing${\displaystyle a=(a_{ij})}$ as a function of${\displaystyle (x_1,\dots ,x_n)}$ valued in the space of${\displaystyle n\times n}$symmetric matrices, alleigenvalues are greater than some positive constant: that is, there is a positive numberθ such that$${\displaystyle \sum _{i=1}^n\sum _{j=1}^n{a}_{ij}(x_1,\dots ,x_n){\xi }_i{\xi }_j\ge \theta ({\xi }_1^2+\cdots +{\xi }_n^2)}$$for every point${\displaystyle (x_1,\dots ,x_n)}$ in the domain and all real numbers${\displaystyle {\xi }_1,\dots ,{\xi }_n}$.^[1][2]

The simplest example of a second-order linear elliptic PDE is theLaplace equation, in which the coefficients are the constant functions${\displaystyle a_{ij}=0}$ for${\displaystyle i\ne j}$,${\displaystyle a_{ii}=1}$, and${\displaystyle b_i=c=f=0}$. ThePoisson equation is a slightly more general second-order linear elliptic PDE, in whichf is not required to vanish. For both of these equations, the ellipticity constantθ can be taken to be1.

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.^[3]

Ellipticity can also be formulated for more general classes of equations. For the most general second-order PDE, which is of the form

${\displaystyle F(D^{2}u,Du,u,x_1,\dots ,x_n)=0}$

for some given functionF,ellipticity is defined bylinearizing the equation and applying the above linear definition. Since linearization is done at a particular function${\displaystyle u}$, 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 simplestMonge–Ampère equation involves thedeterminant of theHessian matrix of the unknown function:

${\displaystyle det{D}^{2}u=f.}$

As follows fromJacobi's formula for the derivative of a determinant, this equation is elliptic if${\displaystyle f}$ is a positive function and solutions satisfy the constraint of beinguniformly convex.^[4]

There are also higher-order elliptic PDE, the simplest example being the fourth-orderbiharmonic equation.^[5] Even more generally, there is an important class ofelliptic systems which consist of coupled partial differential equations for multiple unknown functions.^[6] For example, theCauchy–Riemann equations fromcomplex analysis can be viewed as a first-order elliptic system for a pair of two-variable real functions.^[7]

Moreover, the class of elliptic PDE (of any order, including systems) is subject to various notions ofweak solutions, i.e., reformulating the equations in a way that allows for solutions with various irregularities (e.g.non-differentiability,singularities ordiscontinuities), so as to model non-smooth physical phenomena.^[8] Such solutions are also important invariational calculus, where thedirect method often produces weak solutions for elliptic systems ofEuler equations.^[9]

Consider a second-order elliptic partial differential equation

${\displaystyle A(x,y)u_{xx}+2B(x,y)u_{xy}+C(x,y)u_{yy}+f(u_x,u_y,u,x,y)=0}$

for a two-variable function${\displaystyle u=u(x,y)}$. 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 aquasilinear equation.

Acanonical form asks for a transformation${\displaystyle (w,z)=(w(x,y),z(x,y))}$ of the${\displaystyle (x,y)}$ domain so that, whenu is viewed as a function ofw andz, the above equation takes the form

${\displaystyle u_{ww}+u_{zz}+F(u_w,u_z,u,w,z)=0}$

for some new functionF. The existence of such a transformation can be establishedlocally ifA,B, andC arereal-analytic functions and, with more elaborate work, even if they are onlycontinuously differentiable. Locality means that the necessary coordinate transformations may fail to be defined on the entire domain ofu, only in some small region surrounding any particular point of the domain.^[10]

Formally establishing the existence of such transformations uses the existence of solutions to theBeltrami equation. From the perspective ofdifferential geometry, the existence of a canonical form is equivalent to the existence ofisothermal coordinates for the associatedRiemannian metric

${\displaystyle A(x,y)d{x}^2+2B(x,y)dxdy+C(x,y)d{y}^2}$

on the domain. (The ellipticity condition for the PDE, namely the positivity ofAC –B², is what ensures that either this tensor or its negation is indeed a Riemannian metric.)

For second-order quasilinear elliptic partial differential equations inmore than two variables, a canonical form doesnot 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.^[11]

For the general second-order linear PDE,characteristics are defined as thenull directions for the associated tensor^[12]

${\displaystyle \sum _{i=1}^n\sum _{j=1}^n{a}_{i,j}(x_1,\dots ,x_n)d{x}^{i}d{x}^j,}$

called theprincipal symbol. Using the technology of thewave front set, characteristics are significant in understanding how irregular points off propagate to the solutionu of the PDE. Informally, the wave front set of a function consists of the points of non-smoothness, in addition to the directions infrequency 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 linearelliptic operatorP with smooth coefficients, the principal symbol is aRiemannian metric and there are no real characteristic directions. According to the previous paragraph, it follows that the wave front set of a solutionu coincides exactly with that ofPu =f. This sets up a basicregularity theorem, which says that iff is smooth (so that its wave front set is empty) then the solutionu is smooth as well. More generally, the points whereu fails to be smooth coincide with the points wheref is not smooth.^[15] Thisregularity 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 ofelliptic PDEs are naturally associated with time-independent solutions ofparabolic PDEs orhyperbolic PDEs. For example, a time-independent solution of theheat equation solvesLaplace's equation. That is, if parabolic and hyperbolic PDEs are associated with modelingdynamical systems then the solutions of elliptic PDEs are associated withsteady states. 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 tolinear elliptic equations; moreover, the regularity theory for nonlinear elliptic equations is much more subtle, with solutions not always being smooth.

• Elliptic boundary value problem
• Elliptic operator
• Hyperbolic partial differential equation
• Parabolic partial differential equation
• Maximum principle (property of solutions)
• Sobolev space

• "Elliptic partial differential equation",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• "Elliptic partial differential equation, numerical methods",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Weisstein, Eric W."Elliptic Partial Differential Equation".MathWorld.

• Partial differential equations
• Ellipses

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