A second-order partial differential equation, i.e., one of the form A u_(x x) + 2B u_(x y) + C u_(y y) + D u_x + E u_y + F = 0, is called elliptic if the matrix Z congruent [A | B B | C] is positive definite. Elliptic partial differential equations have applications in almost all areas of mathematics, from harmonic analysis to geometry to Lie theory, as well as numerous applications in physics. As with a general PDE, elliptic PDE may have non-constant coefficients and be non-linear. Despite this variety, the elliptic equations have a well-developed theory.

harmonic function | harmonic map | hyperbolic partial differential equation | Laplace's equation | minimal surface | parabolic partial differential equation | partial differential equation