Subharmonic function

In mathematics , subharmonic and superharmonic functions denote important classes of functions that have their applications in the theory of partial differential equations , function theory and potential theory .

Subharmonic functions are connected to convex functions of a variable as follows: If the graph of a convex function and a straight line intersect at two points, the graph of the convex function is below the straight line between these two points. In the same way, the values ​​of a subharmonic function inside a sphere are no greater than those of a harmonic function if this is true for the edge of the sphere. These properties can be used to define subharmonic functions.

Superharmonic functions can be defined in the same way, replacing "not greater" with "not less". Alternatively, a function can be defined as superharmonic when is subharmonic. Therefore, every property of subharmonic functions can easily be transferred to superharmonic functions. ${\displaystyle f}$ ${\displaystyle -f}$

Formal definition

Let be a subset of Euclidean space and be ${\displaystyle G}$ ${\displaystyle {\mathbb {R} }^{n}}$

${\displaystyle \varphi \colon G\to {\mathbb {R} }\cup \{-\infty \}}$

a semi-permanent function . Then it is subharmonic if for every closed sphere with center and radius out and for every real-valued, continuous function on which is harmonic in and fulfilled for all on the boundary of , always holds for all . ${\displaystyle \varphi }$ ${\displaystyle {\overline {B(x,r)}}}$ ${\displaystyle x}$ ${\displaystyle r}$ ${\displaystyle G}$ ${\displaystyle h}$ ${\displaystyle {\overline {B(x,r)}}}$ ${\displaystyle B(x,r)}$ ${\displaystyle \varphi (y)\leq h(y)}$ ${\displaystyle y}$ ${\displaystyle \partial B(x,r)}$ ${\displaystyle B(x,r)}$ ${\displaystyle \varphi (y)\leq h(y)}$ ${\displaystyle y\in B(x,r)}$

Thus the function, which is identical to −∞, is also subharmonic. However, some authors exclude this case by definition.

properties

• An above- semi-continuous function is subharmonic if and only if with holds for each ${\displaystyle \varphi \colon G\to \mathbb {R} \cup \{-\infty \}}$ ${\displaystyle x\in G}$ ${\displaystyle {\overline {B(x,r)}}\subseteq G}$

${\displaystyle \varphi (x)\leq {\frac {1}{|\partial B(x,r)|}}\int _{\partial B(x,r)}\varphi (s)d\eta ,}$

where denotes the surface dimension. This means that a subharmonic function is at no point greater than the arithmetic mean of its values ​​on a circle around this point. ${\displaystyle d\eta }$

• The maximum of a subharmonic function cannot be assumed inside its domain if the function is not constant. This is the so-called maximum principle , which follows directly from the preceding property.
• A function is harmonic if and only if it is both subharmonic and superharmonic.
• If twice continuously differentiable on an open amount of is, then , if and only if subharmonic ${\displaystyle \varphi }$ ${\displaystyle G}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \varphi }$

${\displaystyle \Delta \varphi \geq 0}$in applies ${\displaystyle G}$

where denotes the Laplace operator . ${\displaystyle \Delta }$

Subharmonic functions in the complex number plane

Subharmonic functions are of particular interest in function theory because they are closely related to holomorphic functions .

A real-valued, continuous function of a complex variable (i.e. of two real variables), which is defined on an open set , is subharmonic if and only if holds for every closed circular disc with center and radius ${\displaystyle \varphi }$ ${\displaystyle G\subset \mathbb {C} }$ ${\displaystyle D(z,r)\subset G}$ ${\displaystyle z}$ ${\displaystyle r}$

${\displaystyle \varphi (z)\leq {\frac {1}{2\pi }}\int _{0}^{2\pi }\varphi (z+re^{i\theta })d\theta .}$

If is a holomorphic function then is ${\displaystyle f}$

${\displaystyle \varphi (z)=\log \left|f(z)\right|}$

subharmonic if you set −∞ at the zeros. ${\displaystyle \varphi (z)}$

In the complex number plane, the connection to the convex functions can also be justified by the fact that a subharmonic function in a domain that is constant in the direction of the imaginary axis is convex in the direction of the real axis, and vice versa. ${\displaystyle f}$ ${\displaystyle G\subset \mathbb {C} }$

Stochastics

In the Markov theory, superharmonic functions are used. If the transition operator is then a function is superharmonic if and only if . Instead of superharmonic, the term excessive is also used. ${\displaystyle P}$ ${\displaystyle f}$ ${\displaystyle Pf\leq f}$

The smallest superharmonic or excessive function that dominates the payoff function is the value of the game.

swell

• John B. Conway: Functions of One Complex Variable. 1st volume 2nd edition. Springer-Verlag, New York NY et al. 1978, ISBN 0-387-90328-3 ( Graduate Texts in Mathematics 11).
• Joseph L. Doob : Classical Potential Theory and Its Probabilistic Counterpart. Springer-Verlag, New York NY et al. 1984, ISBN 3-540-90881-1 ( Basic Teachings of Mathematical Sciences 262).
• Steven G. Krantz : Function Theory of Several Complex Variables. 2nd edition, reprinted with corrections. AMS Chelsea Publishing, Providence RI 2001, ISBN 0-8218-2724-3 .