Subharmonic function

In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory.

Intuitively, subharmonic functions are related to convex functions of one variable as follows. 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 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.

"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.

Formal definition

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

:$varphi\; colon\; G\; o\; \{mathbb\{R\; cup\; \{\; -\; infty\; \}$

be an upper semi-continuous function. 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.

Properties

* A function is harmonic if and only if it is both subharmonic and superharmonic.
* If $phi$ is "C"² (twice continuously differentiable) on an open set $G$ in $\{mathbb\{R^n$, then $phi$ is subharmonic if and only if one has:$Delta\; phi\; ge\; 0$ on $G$:where $Delta$ is the Laplacian.
* 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.

ubharmonic functions in the complex plane

Subharmonic functions are of a particular importance in complex analysis, where they are intimately connected to holomorphic functions.

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

:$varphi(z)\; leq\; frac\{1\}\{2pi\}\; int\_0^\{2pi\}\; varphi(z+\; r\; e^\{i\; heta\})\; d\; heta.$

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.

If $f$ is a holomorphic function, then :$varphi(z)\; =\; log\; left|\; f(z)\; ight|$ 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 $Gsubsetmathbb\{C\}$ that is constant in the imaginary direction is convex in the real direction and vice versa.

Subharmonic functions on Riemannian manifolds

Subharmonic functions can be defined on an arbitrary Riemannian manifold.

"Definition:" Let "M" be a Riemannian manifold, and $f:;\; M\; mapsto\; \{Bbb\; R\}$ an upper semicontinuous function. Assume that for any open subset $Usubset\; M$, and any harmonic function "f₁" on "U", such that $f\_1leq\; f$ on the boundary of "U", the inequality $f\_1leq\; f$ holds on all "U". Then "f" is called "subharmonic".

This definition is equivalent to one given above. Also, for twice differentiable functions, subharmonicity is equivalent to the inequality $Delta\; fgeq\; 0$, where $Delta$ is the usual Laplacian. [citation
author = Greene, R. E.
year = 1974
title = Integrals of subharmonic functions on manifolds of nonnegative curvature
journal = Inventiones mathematicae
volume = 27
pages = 265–298
doi = 10.1007/BF01425500, MathSciNet | id = 0382723]

ee also

* Plurisubharmonic function — generalization to several complex variables
* Classical fine topology

References

*
*
*cite book
last = Doob
first = Joseph Leo
authorlink = Joseph Leo Doob
title = Classical Potential Theory and Its Probabilistic Counterpart
publisher = Springer-Verlag
location = Berlin Heidelberg New York
year = 1984
isbn = 3-540-41206-9 ----

Notes