Elliptic Partial Differential Equation

A second-order Partial Differential Equation, i.e., one of the form

$$Au_{xx} + 2B u_{xy} + C u_{yy} + Du_x + Eu_y + F = 0,$$ (1)

is called elliptic if the Matrix

$${\hbox{\sf Z}} \equiv \left[{\matrix{A & B\cr B & C\cr}}\right]$$ (2)

is Positive Definite. Laplace's Equation and Poisson's Equation are examples of elliptic partial differential equations. For an elliptic partial differential equation, Boundary Conditions are used to give the constraint $u(x,y) = g(x,y)$ on $\partial \Omega$, where

$$u_{xx} + u_{yy} = f(u_x,u_y,u,x,y)$$ (3)

holds in $\Omega$.

See also Hyperbolic Partial Differential Equation, Parabolic Partial Differential Equation, Partial Differential Equation