Harmonic function

In calculus, a real-valued, twice continuously differentiable function is called harmonic, if the application of the Laplace operator yields the function is zero, so the function is a solution of the Laplace equation. The concept of harmonic functions can also be applied to distributions and differential forms.

Definition

Be an open subset. A function is said to blend in, if it is twice continuously differentiable and for all

Applies. This denotes the Laplace operator.

Mean value property

The most important property of harmonic functions is the mean value property, which is equivalent to the definition:

A continuous function is precisely to be harmonious if it satisfies the mean value property, that is, if

For all balls. Herein, the surface measure of the unit -dimensional sphere.

Other properties

The other properties of harmonic functions are largely consequences of the mean value property.

• Maximum Principle: The interior of a coherent definition area a harmonic function at its maximum and its minimum never assumes, unless it is constant. If the function has also a continuous extension to the financial statements, the maximum and minimum on the boundary are assumed.
• Smoothness: A harmonic function is infinitely differentiable. This is remarkable especially in the formulation by means of the averaging property where only the continuity of the function is required.
• Estimation of the derivatives: Be harmonically in. Then for the derivatives wherein the volume of the dimensional unit sphere respectively.
• Analyticity: For the estimation of the derivatives follows that any harmonic function can be expanded in a convergent Taylor series.
• Liouville's theorem: A bounded harmonic function is constant.
• Harnack 's inequality: For any coherent, open and relatively compact subset, there is a constant that depends only on the field, so that for each in harmonic and non-negative function applies.
• The harmonic functions can be regarded as real parts of analytic functions of a complex variable In the special case of a simply connected domain.
• Each harmonic function is also a biharmonic function.

Example

The basic solution

Is a harmonic function on which the measure of the unit sphere referred to. Provided with this normalization makes the basic solution a fundamental role in the theory of Poisson 's equation.