Subharmonic function

In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Potential theory may be defined as the study of Harmonic functions Definition and comments The term "potential theory" arises from the fact that

Intuitively, subharmonic functions are related to convex functions of one variable as follows. In Mathematics, a real-valued function f defined on an interval (or on any Convex subset of some Vector space) is called convex If the graph of a convex function and a line intersect at two points, then the graph of the convex function is below the line between those points. In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x) In the same way, if the values of a subharmonic function are no larger than the values of a harmonic function on the boundary of a ball, then the values of the subharmonic function are no larger than the values of the harmonic function also inside the ball. In Mathematics, Mathematical physics and the theory of Stochastic processes a harmonic function is a twice continuously differentiable function In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric

Superharmonic functions can be defined by the same description, only replacing "no larger" with "no smaller". Alternatively, a superharmonic function is just the negative of a subharmonic function, and for this reason any property of subharmonic functions can be easily transferred to superharmonic functions. In mathematics the additive inverse, or opposite, of a Number n is the number that when added to n, yields zero.

1 Formal definition
2 Properties
3 Subharmonic functions in the complex plane
4 Subharmonic functions on Riemannian manifolds
5 See also
6 References
7 Notes

Formal definition

Formally, the definition can be stated as follows. Let $G$ be a subset of the Euclidean space ${\mathbb{R}}^n$ and let

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

be an upper semi-continuous function. For the notion of upper or lower semicontinuous Multivalued function see Hemicontinuity In Mathematical analysis, semi-continuity Then, $\varphi$ is called subharmonic if for any closed ball $\overline{B(x,r)}$ of centre $x$ and radius $r$ contained in $G$ and every real-valued continuous function $h$ on $\overline{B(x,r)}$ that is harmonic in $B (x, r)$ and satisfies $\varphi(x) \leq h(x)$ for all $x$ on the boundary $\partial B(x,r)$ of $B (x, r)$ we have $\varphi(x) \leq h(x)$ for all $x \in B(x,r).$

Note that by the above, the function which is identically −∞ is subharmonic, but some authors exclude this function by definition. In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output For a different notion of boundary related to Manifolds see that article

Properties

A function is harmonic if and only if it is both subharmonic and superharmonic. In Mathematics, Mathematical physics and the theory of Stochastic processes a harmonic function is a twice continuously differentiable function ↔
If $φ$ is C² (twice continuously differentiable) on an open set $G$ in ${\mathbb{R}}^n$ , then $φ$ is subharmonic if and only if one has

$\Delta \phi \ge 0$ on

G

where

Δ

is the Laplacian. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in ↔ In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\ or \nabla^2 and named after

The maximum of a subharmonic function cannot be achieved in the interior of its domain unless the function is constant, this is the so-called maximum principle. In Mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum or smallest value (minimum that In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S " This article describes the maximum principle in mathematics For the maximum principle in optimal control theory see Pontryagin's minimum principle.

Subharmonic functions in the complex plane

Subharmonic functions are of a particular importance in complex analysis, where they are intimately connected to holomorphic functions. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane

One can show that a real-valued, continuous function $\varphi$ of a complex variable (that is, of two real variables) defined on a set $G\subset \mathbb{C}$ is subharmonic if and only if for any closed disc $D(z,r) \subset G$ of center $z$ and radius $r$ one has

$\varphi(z) \leq \frac{1}{2\pi} \int_0^{2\pi} \varphi(z+ r e^{i\theta}) d\theta.$

Intuitively, this means that a subharmonic function is at any point no greater than the average of the values in a circle around that point, a fact which can be used to derive the maximum principle. In Mathematics and Statistics, the arithmetic Mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided This article describes the maximum principle in mathematics For the maximum principle in optimal control theory see Pontryagin's minimum principle.

If $f$ is a holomorphic function, then

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

is a subharmonic function if we define the value of $\varphi(z)$ at the zeros of $f$ to be −∞.

In the context of the complex plane, the connection to the convex functions can be realized as well by the fact that a subharmonic function $f$ on a domain $G\subset\mathbb{C}$ that is constant in the imaginary direction is convex in the real direction and vice versa. In Mathematics, a real-valued function f defined on an interval (or on any Convex subset of some Vector space) is called convex

Subharmonic functions on Riemannian manifolds

Subharmonic functions can be defined on an arbitrary Riemannian manifold. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M

Definition: Let M be a Riemannian manifold, and $f:\; M \mapsto {\Bbb R}$ an upper semicontinuous function. For the notion of upper or lower semicontinuous Multivalued function see Hemicontinuity In Mathematical analysis, semi-continuity Assume that for any open subset $U\subset M$ , and any harmonic function f₁ on U, such that $f_1\leq f$ on the boundary of U, the inequality $f_1\leq f$ holds on all U. In Mathematics, Mathematical physics and the theory of Stochastic processes a harmonic function is a twice continuously differentiable function Then f is called subharmonic.

This definition is equivalent to one given above. Also, for twice differentiable functions, subharmonicity is equivalent to the inequality $\Delta f\geq 0$ , where $Δ$ is the usual Laplacian. In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\ or \nabla^2 and named after ^[1]

References

John B. In Mathematics, plurisubharmonic functions form an important class of functions used in Complex analysis. The theory of functions of several complex variables is the branch of Mathematics dealing with functions f ( z1 z2 In Mathematics, in the field of Potential theory, the fine topology is a natural topology for setting the study of Subharmonic functions In the earliest Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.
Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Doob, Joseph Leo (1984). Joseph Leo Doob ( February 27, 1910 - June 7, 2004) was an American Mathematician, specializing in analysis Classical Potential Theory and Its Probabilistic Counterpart. Berlin Heidelberg New York: Springer-Verlag. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic ISBN 3-540-41206-9.

Notes

^ Greene, R. E. ; Wu, H. Integrals of subharmonic functions on manifolds of nonnegative curvature. Invent. Math. 27 (1974), 265--298. MR MR0382723

This article incorporates material from Subharmonic and superharmonic functions on PlanetMath, which is licensed under the GFDL. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many PlanetMath is a free, collaborative online Mathematics Encyclopedia.

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