Laplace-Operator

The Laplacian is a mathematical operator that was first introduced by Pierre -Simon Laplace. There is a linear differential operator within the multidimensional analysis. It is usually quoted by the character that is the uppercase letter delta of the Greek alphabet.

The Laplacian is found in many differential equations that describe the behavior of physical fields. Examples are the Poisson equation of electrostatics and the diffusion equation for the heat conduction. Often the Laplacian is used to calculate the distribution of gravitational fields. The scalar Laplace operator is to be distinguished from the vector Laplace operator, which occurs in the wave equations for electromagnetic fields.

• 4.1 Definition
• 4.2 Green's function

• 5.1 D' Alembert operator
• 5.2 Generalized Laplacian

Definition

The Laplace operator assigns a twice differentiable scalar field, the divergence of its gradient to

Or listed with the nabla operator

Being mainly in the English-speaking world for the Laplace operator often to find the spelling listed on the right.

Since the divergence operator and the gradient operator are independent of the chosen coordinate system, and the Laplace operator is independent of the chosen coordinate system. The representation of the Laplacian in different coordinate systems results with the chain rule from the coordinate transformation.

In n-dimensional Euclidean space is obtained in Cartesian coordinates

In one dimension is the Laplace operator, thus reducing the second derivative:

The Laplacian of a function can be represented as a trace of their Hessian matrix:

Representation

In two dimensions,

For a function in Cartesian coordinates provides the application of the Laplacian

In polar coordinates results

Or

In three dimensions

For a function with three variables results in Cartesian coordinates

In cylindrical coordinates results

And in spherical coordinates

This representation is also used in out stapled shape with change of the first and second term. The first ( radial ) term can be written in three equivalent forms:

These representations of the Laplace operator in cylindrical and spherical coordinates are only valid for the scalar Laplace operator. Further terms still need to be considered for the Laplace operator acting on vector-valued functions ( vectorial Laplace operator ).

In curvilinear orthogonal coordinates

In contrast, applies with arbitrary curvilinear orthogonal coordinates, for example, in spherical polar coordinates, cylindrical coordinates, elliptical coordinates

Which is (= 1 for i = k, = 0 otherwise ), because

So where not, but the sizes have the physical dimension of " length", a more general relation for the Laplace operator, bearing in mind that are not constant, but of, and may depend on:

Two terms are defined by the points, ..., indicated that emerge from the prescribed term by cyclic permutation according to the scheme 1 → 2, 2 → 3, 3 → 1. For more general coordinate the Laplace Beltrami relationship applies ( see below).

Properties

The Laplacian is a linear operator, that is, are twice and differentiable functions and constants, the following applies

As for other linear differential operators also applies a generalized product rule for the Laplace operator. This is

Where two twice continuously differentiable functions with and the Euclidean standard scalar is.

The Laplacian is rotationally symmetric, that is, is a twice -differentiable function, and a rotation, the following applies

Where " " denotes the concatenation of images.

The main symbol of the Laplacian is. So he is an elliptic differential operator of second order. It follows that he is a Fredholm operator and follows by means of the set of Atkinson that he is right and linksinvertierbar modulo a compact operator.

Is the Laplace operator

On the Schwartz space is essentially self-adjoint. He therefore has a seclusion

To a self-adjoint operator on the Sobolev space. This operator is also still positive, its spectrum is thus located on the non- negative real axis, that is

The eigenvalue equation

Of the Laplacian is called Helmholtz equation. Is a bounded domain and the Sobolev space with the boundary values ​​. Then the eigenfunctions of the Laplace operator form a complete orthonormal system and its spectrum consists of a purely discrete, real point spectrum which can have only one accumulation point. This follows from the spectral theorem for self-adjoint elliptic differential operators.

Clearly there Δƒ ( p) for a function ƒ at a point p on how changed the mean value of ƒ over concentric spherical shells around p with increasing sphere radius over ƒ (p).

Poisson and Laplace equation

Definition

The Laplacian occurs in a number of important differential equations. The homogeneous differential equation

Is called the Laplace equation and twice continuously differentiable solutions of this equation are called harmonic functions. The corresponding inhomogeneous equation

Is Poisson's equation.

Green's function

The Green function of the Laplace operator satisfies the Poisson equation

With the delta function on the right side. For this reason, the Green function is the fundamental solution of the Poisson equation. The Green's function is dependent on the number of spatial dimensions.

In three dimensions it is:

This Green's function is needed in electrodynamics as a tool to solve boundary value problems.

In two dimensions is the Green's function:

Generalizations

D' Alembert operator

The Laplacian results together with the second time derivative of the d' Alembert operator:

This operator may be considered as a generalization of the Laplacian at the Minkowski space.

Generalized Laplace operator

For the Laplace operator, which was originally always understood as an operator of Euclidean space, there was the possibility of generalization to curved surfaces and Riemannian or pseudo- Riemannian manifolds with the formulation of the Riemannian geometry. This general operator will be referred to as a generalized Laplace operator.

Discrete Laplace operator and image processing

In image processing, the Laplacian is used for edge detection. An edge appears as a zero crossing of the second derivative of the signal. A discrete signal gn and GNM the Laplacian operator is applied by a convolution. Use can use the following simple convolution masks:

For the 2D filter, there is a second variant, which additionally takes into account diagonal edges:

This convolution mask is obtained by discretization of the differential quotients.