Harmonic function

A harmonic function defined on a circular ring .

In analysis , a real-valued function that is twice continuously differentiable is called harmonic if the application of the Laplace operator to the function results in zero, i.e. 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 called harmonic in if it is twice continuously differentiable and for all ${\displaystyle U\subseteq \mathbb {R} ^{n}}$ ${\displaystyle f\colon U\rightarrow \mathbb {R} }$ ${\displaystyle U}$ ${\displaystyle x\in U}$

${\displaystyle \Delta f(x)=0}$

applies. It denotes the Laplace operator . ${\displaystyle \Delta ={\tfrac {\partial ^{2}}{\partial x_{1}^{2}}}+{\tfrac {\partial ^{2}}{\partial x_{2}^{2}}}+\cdots +{\tfrac {\partial ^{2}}{\partial x_{n}^{2}}}}$

Mean property

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

A continuous function is harmonic if and only if it fulfills the mean value property, that is, if ${\displaystyle f\colon U\rightarrow \mathbb {R} }$

${\displaystyle f(x)={\frac {1}{r^{n-1}s_{n-1}}}\int _{\partial B(x,r)}f(y)\mathrm {d} \sigma (y)}$

for all balls with . Here denotes the area of ​​the -dimensional unit sphere (see Sphere (Mathematics) #Content and Volume ). ${\displaystyle \ B(x,r)}$ ${\displaystyle {\overline {B}}(x,r)\subset U}$ ${\displaystyle s_{n-1}}$ ${\displaystyle n-1}$

Other properties

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

• Maximum principle : Inside a coherent domain of definition , a harmonic function never assumes its maximum and its minimum, except when it is constant. If the function also has a continuous continuation to the conclusion , the maximum and minimum are assumed on the edge . ${\displaystyle U}$ ${\displaystyle {\overline {U}}}$ ${\displaystyle \partial U}$
• Smoothness : A harmonic function can be differentiated any number of times. This is particularly noticeable in the formulation with the aid of the mean value property, where only the continuity of the function is assumed.
• Estimation of the derivatives: Be harmonious in . Then applies for the derivatives where denotes the volume of the -dimensional unit sphere . ${\displaystyle f}$ ${\displaystyle U}$
  ${\displaystyle \left|D^{\alpha }f(x)\right|\leq {\frac {\left(2^{n+1}n|\alpha |\right)^{|\alpha |}}{v_{n}}}\left\|f\right\|_{L^{1}(B(x,r))},}$
  ${\displaystyle v_{n}}$ ${\displaystyle n}$
• Analyticity : From the estimation of the derivatives it follows that every harmonic function can be expanded into a convergent Taylor series .
• Liouville's theorem : A bounded harmonic function is constant. ${\displaystyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} }$
• Harnack's inequality : for every connected, open and relatively compact subset there is a constant that only depends on the area , so that for every in harmonic and nonnegative function . ${\displaystyle V\subset \subset U}$ ${\displaystyle C\geq 0}$ ${\displaystyle V}$ ${\displaystyle U}$ ${\displaystyle f}$
  ${\displaystyle \sup _{V}f\leq C\inf _{V}f}$
  
• In the special case for a simply connected area , the harmonic functions can be understood as real parts of analytical functions of a complex variable. ${\displaystyle n=2}$ ${\displaystyle U\subset \mathbb {R} ^{2}\cong \mathbb {C} }$
• Every harmonic function is also a biharmonic function .

example

The basic solution

${\displaystyle S(x):=\left\{{\begin{array}{ll}-{\frac {1}{2\pi }}\ln |x|\ ,&n=2\ ,\\{\frac {1}{(n-2)\omega _{n}}}{\frac {1}{\|x\|^{n-2}}}\ ,&n\geq 3\ ,\\\end{array}}\right.}$

is an on harmonic function, in which the measure of the unit sphere im denotes. If this normalization is provided, the basic solution plays a fundamental role in the theory of the Poisson equation . ${\displaystyle \mathbb {R} ^{n}\setminus \{0\}}$ ${\displaystyle \omega _{n}}$ ${\displaystyle \mathbb {R} ^{n}}$

Generalizations

Polyharmonic functions are continuous solutions of the differential equation up to the 2mth order of the derivative:

${\displaystyle {\Delta }^{m}f=0}$

For m = 2 ( biharmonic function ) the differential equation appears in the theory of elastic plates ( Gustav Kirchhoff ).

literature

• Lawrence C. Evans : Partial Differential Equations. Reprinted with corrections. American Mathematical Society, Providence RI 2002, ISBN 0-8218-0772-2 ( Graduate studies in mathematics 19).