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subharmonic and superharmonic functions
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(Definition)
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First let's look at the most general definition.
Definition 1 Let $G \subset {\mathbb{R}}^n$ and let $\varphi \colon G \to {\mathbb{R}} \cup \{ - \infty \}$ be an upper semi-continuous function, then $\varphi$ is subharmonic if for every $x \in G$ and $r > 0$ such that $\overline{B(x,r)} \subset G$ (the closure of the open ball of radius $r$ around $x$ is still 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 $x \in \partial B(x,r)$ ( boundary of $B(x,r)$ we have that $\varphi(x) \leq h(x)$ holds for all $x \in B(x,r)$
Note that by the above, the function which is identically $- \infty$ is subharmonic, but some authors exclude this function by definition. We can define superharmonic functions in a similar fashion to get that $\varphi$ is superharmonic if and only if $-\varphi$ is subharmonic.
If we restrict our domain to the complex plane we can get the following definition.
Definition 2 Let $G \subset {\mathbb{C}}$ be a region and let $\varphi \colon G \to {\mathbb{R}}$ be a continuous function. $\varphi$ is said to be subharmonic if whenever $D(z,r) \subset G$ (where $D(z,r)$ is a closed disc around $z$ of radius $r$ we have \begin{equation*} \varphi(z) \leq \frac{1}{2\pi} \int_0^{2\pi} \varphi(z+ r e^{i\theta}) d\theta , \end{equation*}and $\varphi$ is said to be
superharmonic if whenever $D(z,r) \subset G$ we have \begin{equation*} \varphi(z) \geq \frac{1}{2\pi} \int_0^{2\pi} \varphi(z+ r e^{i\theta}) d\theta . \end{equation*}
Intuitively what this means is that a subharmonic function is at any point no greater than the average of the values in a circle around that point. This implies that a non-constant subharmonic function does not achieve its maximum in a region $G$ (it would achieve it at the boundary if it is continuous there). Similarly for a superharmonic function, but then a non-constant superharmonic function does not achieve its minumum in $G$ It is also easy to see that
$\varphi$ is subharmonic if and only if $-\varphi$ is superharmonic.
Note that when equality always holds in the above equation then $\varphi$ would in fact be a harmonic function. That is, when $\varphi$ is both subharmonic and superharmonic, then $\varphi$ is harmonic.
It is possible to relax the continuity statement above to take $\varphi$ only upper semi-continuous in the subharmonic case and lower semi-continuous in the superharmonic case. The integral will then however need to be the Lebesgue integral rather than the Riemann integral which may not be defined for such a function. Another thing to note here is that we may take ${\mathbb{R}}^2$ instead of ${\mathbb{C}}$ since we never did use complex multiplication. In that case however we must rewrite the expression $z + r e^{i\theta}$ in terms of the real and imaginary parts to get an expression in ${\mathbb{R}}^2$
It is also possible generalize the range of the functions as well. A subharmonic function could have a range of ${\mathbb{R}} \cup \{ -\infty \}$ and a superharmonic function could have a range of ${\mathbb{R}} \cup \{ \infty \}$ With this generalization, if $f$ is a holomorphic function then $\varphi(z) := \log \lvert f(z) \rvert$ is a subharmonic function if we define the value of $\varphi(z)$ at the zeros of $f$ as $-\infty$ Again it is important to note that with this generalization we again must use the Lebesgue
integral.
- 1
- John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.
- 2
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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"subharmonic and superharmonic functions" is owned by jirka.
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See Also: harmonic function
| Also defines: |
subharmonic, subharmonic function, superharmonic, superharmonic function |
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Cross-references: Lebesgue integral, holomorphic function, range, imaginary parts, expression, complex multiplication, Riemann integral, integral, lower semi-continuous, harmonic function, equation, equality, easy to see, implies, circle, average, point, disc, closed, region, complex plane, domain, similar, boundary, harmonic, continuous function, real, radius, open ball, closure, function, upper semi-continuous
There are 4 references to this entry.
This is version 9 of subharmonic and superharmonic functions, born on 2004-04-22, modified 2005-03-07.
Object id is 5796, canonical name is SubharmonicAndSuperharmonicFunctions.
Accessed 8962 times total.
Classification:
| AMS MSC: | 31B05 (Potential theory :: Higher-dimensional theory :: Harmonic, subharmonic, superharmonic functions) | | | 31A05 (Potential theory :: Two-dimensional theory :: Harmonic, subharmonic, superharmonic functions) | | | 31C05 (Potential theory :: Other generalizations :: Harmonic, subharmonic, superharmonic functions) |
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Pending Errata and Addenda
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