Elliptic partial differential equation
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form where , , , , , , and are functions of and and where formula_2, formula_3 and similarly for formula_4. A PDE written in this form is elliptic if with this naming convention inspired by the equation for a planar ellipse. The simplest nontrivial examples of elliptic PDE's are the Laplace equation, formula_6, and the Poisson equation, formula_7 In a sense, any other elliptic PDE in two variables can be considered to  be a generalization of one of these equations, as it can always be put into the canonical form through a change of variables. Qualitative behavior. Elliptic equations have no real characteristic curves, curves along which it is not possible to eliminate at least one second derivative of formula_9 from the conditions of the Cauchy problem. 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 formula_10 by setting formula_11. This means that Laplace's equation describes a steady state of the heat equation. 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. Derivation of canonical form. We derive the canonical form for elliptic equations in two variables, formula_12. If  formula_15, applying the chain rule once gives a second application gives We can replace our PDE in x and y with an equivalent equation in formula_21 and formula_22 where To transform our PDE into the desired canonical form, we seek formula_21 and formula_22 such that formula_29 and formula_30. This gives us the system of equations Adding formula_33 times the second equation to the first and setting formula_34 gives the quadratic equation Since the discriminant formula_36, this equation has two distinct solutions, which are complex conjugates. Choosing either solution, we can solve for formula_38, and recover formula_21 and formula_22 with the  transformations formula_41 and formula_42. Since formula_22 and formula_21 will satisfy formula_45 and formula_30, so with a change of variables from x and y to formula_22 and formula_21 will transform the PDE into the canonical form as desired. In higher dimensions. A general second-order partial differential equation in variables takes the form This equation is considered elliptic if there are no characteristic surfaces, i.e. surfaces along which it is not possible to eliminate at least one second derivative of from the conditions of the Cauchy problem. Unlike the two-dimensional case, this equation cannot in general be reduced to  a simple canonical form.