INDEX

Introduction

Laplace equation in Cartesian coordinates

Laplace equation in two dimensions

Connection with holomorphic functions

A second-order partial differential equation

Second-order partial differential equation is elliptic

Proof

References

Introduction

Laplace's equation

Laplace's equation is a partial differential equation. Partial differential equations (PDE) are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic.

Laplace's equation is named after Pierre-Simon Laplace who first studied its properties. The solutions of Laplace's equation are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics; because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials. The general theory of solutions to Laplace's equation is known as potential theory. In the study of heat conduction, the Laplace equation is the steady-state heat equation.

Laplace equation in Cartesian coordinates

Given a scalar field , the Laplace equation in Cartesian coordinates x, y, and z, such that

This is often written as

or

where is the divergence, and is the gradient, or

where Δ is the Laplace operator.

Solutions of Laplace's equation are called harmonic functions.

If the right-hand side is specified as a given function, f(x, y, z), i.e., if the whole equation is written as

then it is called "Poisson's equation".

Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. The partial differential operator, or (which may be defined in any number of dimensions), is called the Laplace operator or just the Laplacian.

Laplace equation in two dimensions

The Laplace equation for an unknown function of two variables φ has the form

Solutions of Laplace's equation are called harmonic functions.

Connection with holomorphic functions

Solutions of the Laplace equation in two dimensions are intimately connected with analytic functions of a complex variable (a.k.a. holomorphic functions): the real and imaginary parts of any analytic function are conjugate harmonic functions: they both satisfy the Laplace equation, and their gradients are orthogonal. If f=u+iv, then the Cauchy–Riemann equations state that

and it follows that

Conversely, given any harmonic function in two dimensions, it is the real part of an analytic function, at least locally. Details are given in Laplace equation.

A second-order partial differential equation

Assuming uxy = uyx, the general second-order PDE in two independent variables has the form

where the coefficients A, B, C etc. may depend upon x and y. This form is analogous to the equation for a conic section:

More precisely, replacing by X, and likewise for other variables (formally this is done by a Fourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with the top degree (a homogeneous polynomial, here a quadratic form) being most significant for the classification.

Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant B2 − 4AC, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by B2 − AC, due to the convention of the xy term being 2B rather than B; formally, the discriminant (of the associated quadratic form) is (2B)2 − 4AC = 4(B2 − AC), with the factor of 4 dropped for simplicity.

1.  : Solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where x<0.

2.  : Equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where x=0.

3.  : Hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where x>0.

If there are n independent variables x1, x2 , ..., xn, a general linear partial differential equation of second order has the form

The classification depends upon the signature of the eigenvalues of the coefficient matrix.

1. Elliptic: The eigenvalues are all positive or all negative.2. Parabolic: The eigenvalues are all positive or all negative, save one that is zero.3. Hyperbolic: There is only one negative eigenvalue and all the rest are positive, or

there is only one positive eigenvalue and all the rest are negative.

4. Ultrahyperbolic: There is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues. There is only limited theory for ultrahyperbolic equations.

Second-order partial differential equation is elliptic

A second-order partial differential equation, i.e., one of the form

is called elliptic if the matrix

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.

The basic example of an elliptic partial differential equation is Laplace's equation

in -dimensional Euclidean space, where the Laplacian is defined by

Other examples of elliptic equations include the nonhomogeneous Poisson's equation

and the non-linear minimal surface equation.

Proof

A steady-state temperature distribution in a plate with a given temperature distribution on its boundary S is described by the Laplace equation uxx + uyy = 0 with the boundary condition u|S = f(x, y),where u(x, y) is the temperature at a position (x, y) and f(x, y) is the given temperature distribution on the boundary.

This equation follows from the two-dimensional heat equation ut = v (uxx + uyy) under steady-state conditions,

u/ t = 0.The Laplace equation has A = C = 1, B = 0, so that B2−AC = −1. This is an elliptic equation.

References

www.google.com

www.wikipedia.com

www.handouts.maths.com

L.C. Evans, Partial Differential Equations.

G. Petrovsky, Partial Differential Equations,

A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers

and Scientists

A. Sommerfeld, Partial Differential Equations in Physics