This result can be used to prove the maximum principle for harmonic functions.

By the maximum principle, u is the only such harmonic function on D.

He also made contributions to the theory of minimal surfaces and harmonic functions.

Like for minimal surfaces, there exist a close link to harmonic functions.

In mathematics and mathematical physics, potential theory is the study of harmonic functions.

The above chords, despite their differences, share the same harmonic function and can be used interchangeably.

Harmonic functions are the classical example to which the strong maximum principle applies.

An important topic in potential theory is the study of the local behavior of harmonic functions.

This definition ties minimal surfaces to harmonic functions and potential theory.

There are results which describe the local structure of level sets of harmonic functions.

The quantity measures how far off the temperature is from satisfying the mean value property of harmonic functions.

It is a work that barely grasps onto tonality and harmonic function.

The principle also adapts to apply to harmonic functions.

Suppose is a harmonic function into the Riemann sphere.

Viscosity solutions to the equation are also known as infinity harmonic functions.

A defining property of the usual -harmonic functions is the mean value property.

Several different normalizations are in common use for the Laplace spherical harmonic functions.

This condition guarantees that the maximum principle will hold, although other properties of harmonic functions may fail.

This equation states that hydraulic head is a harmonic function, and has many analogs in other fields.

Any harmonic function is biharmonic, but the converse is not always true.

Such a solution is called a harmonic function and such solutions are the topic of study in potential theory.

The weak maximum principle for harmonic functions is a simple consequence of facts from calculus.

Examples of harmonic functions of three variables are given in the table below with :

In fact, harmonic functions are real analytic.

The following principle of removal of singularities holds for harmonic functions.