Subharmonic function

In mathematics refer to sub-harmonic and super- harmonic functions important classes of functions, which in theory have their applications Partial differential equations, function theory and potential theory.

Subharmonic functions associated to convex functions of one variable as follows: If the graph of a convex function and a straight line intersect at two points is the graph of the convex function below the straight line between these two points. In the same way the values ​​of a sub-harmonic function inside a ball are not larger than that of a harmonic function when this is true for the edge of the sphere. Subharmonic functions can be defined by these properties.

Super harmonic functions can be defined in the same way, which is " not greater than" is replaced by " not less ". Alternatively, a function can be defined as super- harmonic if it is subharmonic. Therefore, any property of subharmonic functions can be easily transferred to super harmonic functions.

Formal definition

Let be a subset of Euclidean space and let

Above a continuous function. Then is subharmonic, if satisfied for every closed ball with center and radius and for any real-valued continuous function on that is harmonious in and for all on the edge of always applies to everyone.

This is also the function that the same - is ∞, subharmonic. However, some authors conclude from this case by definition.

Properties

  • A continuous function above is accurate then subharmonic if for every valid with
  • The maximum of a subharmonic function can not be assumed in the interior of its domain, if the function is not constant. This is the so-called maximum principle which follows directly from the foregoing property.
  • A function is precisely to be harmonious if it is both subharmonic and superharmonic.
  • When is twice continuously differentiable on an open set, then is subharmonic if and only if

Subharmonic functions in the complex plane

Subharmonic functions in the theory of functions of particular interest, since they are closely connected with holomorphic functions.

A real-valued, continuous function of a complex variable (ie of two real variables), which is defined on an open set, if and only subharmonic if for any closed disk with center and radius

If a holomorphic function, then

Subharmonic if one of the zeros on - is ∞.

In the complex plane can be justified to the convex functions also by the fact the connection that a subharmonic function in a field that is constant in the direction of the imaginary axis, is convex in the direction of the real axis, and vice versa.

Stochastics

In the Markov theory of superharmonic functions. Is the transition operator, a function is superharmonic if and only if. Instead superharmonic the term is used excessively.

The smallest superharmonic or excessive function majorized the payoff function, the value of the game.

Swell

  • John B. Conway: Functions of One Complex Variable. Volume 1 2nd edition. Springer - Verlag, New York, NY, inter alia, 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, inter alia, 1984, ISBN 3-540-90881-1 (basic teachings of Mathematical Sciences 262).
  • Steven G. Krantz: Function Theory of Several Complex Variables. Second edition, reprinted with corrections. AMS Chelsea Publishing, Providence RI, 2001, ISBN 0-8218-2724-3.
  • Analysis
  • Function theory
  • Partial Differential Equations
Complex analysis Laplace-Operator
753143
de