Elliptic partial differential equation

In mathematics, an elliptic partial differential equation is a type of second-order partial differential equation (PDE).

Any second-order linear PDE in two variables can be written in the form

Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+Fu+G=0,\,

where $A$ , $B$ , $C$ , $D$ , $E$ , $F$ , and $G$ are functions of $x$ and $y$ and where $u_{x}={\frac {\partial u}{\partial x}}$ , $u_{xy}={\frac {\partial ^{2}u}{\partial x\partial y}}$ and similarly for $u_{xx},u_{y},u_{yy}$ . A PDE written in this form is elliptic if

B^{2}-AC<0,

with this naming convention inspired by the equation for a planar ellipse. Equations with $B^{2}-AC=0$ are termed parabolic while those with $B^{2}-AC>0$ are hyperbolic.

Qualitative behavior

Elliptic equations have no real characteristic curves, curves along which it is not possible to eliminate at least one second derivative of $u$ from the conditions of the Cauchy problem.^[1] Since characteristic curves are the only curves along which solutions to partial differential equations with smooth parameters can have discontinuous derivatives, solutions to elliptic equations cannot have discontinuous derivatives anywhere. This means elliptic equations are well suited to describe equilibrium states, where any discontinuities have already been smoothed out. For instance, we can obtain Laplace's equation from the heat equation $u_{t}=\Delta u$ by setting $u_{t}=0$ . This means that Laplace's equation describes a steady state of the heat equation.^[2]

In parabolic and hyperbolic equations, characteristics describe lines along which information about the initial data travels. Since elliptic equations have no real characteristic curves, there is no meaningful sense of information propagation for elliptic equations. This makes elliptic equations better suited to describe static, rather than dynamic, processes.^[2]

Definition

Elliptic differential equations appear in many different contexts and levels of generality. It is very common to see elliptic PDE which are linear and second-order. For a general linear second-order PDE, the "unknown" function $u$ can be a function of any number $x 1, ..., x n$ of independent variables; the equation is of the form

\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}(x_{1},\ldots ,x_{n}){\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}+\sum _{i=1}^{n}b_{i}(x_{1},\ldots ,x_{n}){\frac {\partial u}{\partial x_{i}}}+c(x_{1},\ldots ,x_{n})u=f(x_{1},\ldots ,x_{n}).

where $a i, j$ , $b i$ , and $c$ are functions defined on the domain subject to the symmetry $a i, j = a j, i$ . This equation is called elliptic if, when $a$ is viewed as a function on the domain valued in the space of $n \times n$ symmetric matrices, all of the eigenvalues are greater than some set positive number. Equivalently, this means that there is a positive number $θ$ such that

\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})

for any point $x 1, ..., x n$ in the domain and any real numbers $ξ 1, ..., ξ n$ .^[3]

The simplest example of a second-order linear elliptic PDE is the Laplace equation, in which $a i, j$ is zero if $i \neq j$ and is one otherwise, and where $b i = c = f = 0$ . The Poisson equation is a slightly more general second-order linear elliptic PDE, in which $f$ is not required to vanish. For both of these equations, the ellipticity constant $θ$ can be taken to be $1$ .

Canonical form

Consider a second-order elliptic partial differential equation

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 $u = u (x, y)$ . This equation is linear in the "leading-order terms" but allows nonlinear expressions involving the function values and their first derivatives; this is sometimes called a quasilinear equation.

A canonical form asks for a transformation $w = w (x, y)$ and $z = z (x, y)$ of the domain so that, when $u$ is viewed as a function of $w$ and $z$ , the above equation takes the form

u_{ww}+u_{zz}+F(u_{w},u_{z},u,w,z)=0

for some new function $F$ . The existence of such a transformation can be established locally if $A$ , $B$ , and $C$ are real-analytic functions 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 $u$ , although they can be established in some small region surrounding any particular point of the domain.^[4]

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

A(x,y)dx^{2}+2B(x,y)\,dx\,dy+C(x,y)dy^{2}

on the domain. 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.^[5]

Generalization

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

F(D^{2}u,Du,u,x_{1},\ldots ,x_{n})=0

for some given function $F$ , ellipticity is defined by linearizing the equation and applying the above linear definition. The Monge–Ampère equation is a standard example of a nonlinear second-order elliptic PDE.^[6] Moreover, the class of elliptic PDE is subject to various notions of weak solutions, in which the usual notions of differentiation which are used to make sense of the above equations are reformulated in novel settings.^[3]^[7]

The terminology elliptic partial differential equation is not used consistently throughout the literature. What is called "elliptic" by some authors is called strictly elliptic or uniformly elliptic by others.^[3]^[6]

References

^ Pinchover, Yehuda; Rubinstein, Jacob (2005). An Introduction to Partial Differential Equations. Cambridge: Cambridge University Press. ISBN 978-0-521-84886-2.
^ ^a ^b Zauderer, Erich (1989). Partial Differential Equations of Applied Mathematics. New York: John Wiley&Sons. ISBN 0-471-61298-7.
^ ^a ^b ^c Evans, Chapter 6
^ Courant and Hilbert. Methods of mathematical physics. Volume II.
^ Spivak, Michael (1979). A comprehensive introduction to differential geometry. Volume V (Second edition of 1975 original ed.). Wilmington, DE: Publish or Perish, Inc. ISBN 0-914098-83-7. MR 0532834.
^ ^a ^b Gilbarg, David; Trudinger, Neil S. (2001). Elliptic partial differential equations of second order. Classics in Mathematics (Revised second edition of the 1977 original ed.). Berlin: Springer-Verlag. doi:10.1007/978-3-642-61798-0. ISBN 3-540-41160-7. MR 1814364. Zbl 1042.35002.
^ Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. doi:10.1090/S0273-0979-1992-00266-5

External links

"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.

[pinchover-1] Pinchover, Yehuda; Rubinstein, Jacob (2005). An Introduction to Partial Differential Equations. Cambridge: Cambridge University Press. ISBN 978-0-521-84886-2.

[Zauderer-2] Zauderer, Erich (1989). Partial Differential Equations of Applied Mathematics. New York: John Wiley&Sons. ISBN 0-471-61298-7.

[evans-3] Evans, Chapter 6

[4] Courant and Hilbert. Methods of mathematical physics. Volume II.

[5] Spivak, Michael (1979). A comprehensive introduction to differential geometry. Volume V (Second edition of 1975 original ed.). Wilmington, DE: Publish or Perish, Inc. ISBN 0-914098-83-7. MR 0532834.

[gilbargtrudinger-6] Gilbarg, David; Trudinger, Neil S. (2001). Elliptic partial differential equations of second order. Classics in Mathematics (Revised second edition of the 1977 original ed.). Berlin: Springer-Verlag. doi:10.1007/978-3-642-61798-0. ISBN 3-540-41160-7. MR 1814364. Zbl 1042.35002.

[7] Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. doi:10.1090/S0273-0979-1992-00266-5

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