subharmonic function
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, subharmonic and superharmonic functions are important classes of functions used extensively in
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
,
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
and
potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
. Intuitively, subharmonic functions are related to
convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poi ...
s 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 A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...
, 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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
\R^n and let \varphi \colon G \to \R \cup \ be an upper semi-continuous function. Then, \varphi is called ''subharmonic'' if for any
closed ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defi ...
\overline of center x and radius r contained in G and every real-valued
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
h on \overline that is
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...
in B(x,r) and satisfies \varphi(y) \leq h(y) for all y on the boundary \partial B(x,r) of B(x,r), we have \varphi(y) \leq h(y) for all y \in B(x,r). Note that by the above, the function which is identically −∞ is subharmonic, but some authors exclude this function by definition. A function u is called ''superharmonic'' if -u is subharmonic.


Properties

* A function is
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
it is both subharmonic and superharmonic. * If \phi is ''C''2 ( twice continuously differentiable) on an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
G in \R^n, then \phi is subharmonic
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
one has \Delta \phi \geq 0 on G, where \Delta is the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
. * The
maximum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
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 the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. ...
. However, the
minimum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
of a subharmonic function can be achieved in the interior of its domain. * Subharmonic functions make a
convex cone In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every . W ...
, that is, a linear combination of subharmonic functions with positive coefficients is also subharmonic. *The pointwise maximum of two subharmonic functions is subharmonic. If the pointwise maximum of a countable number of subharmonic functions is upper semi-continuous, then it is also subharmonic. *The limit of a decreasing sequence of subharmonic functions is subharmonic (or identically equal to -\infty). *Subharmonic functions are not necessarily continuous in the usual topology, however one can introduce the fine topology which makes them continuous.


Examples

If f is
analytic Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic or analytical can also have the following meanings: Chemistry * ...
then \log, f, is subharmonic. More examples can be constructed by using the properties listed above, by taking maxima, convex combinations and limits. In dimension 1, all subharmonic functions can be obtained in this way.


Riesz Representation Theorem

If u is subharmonic in a region D, in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
of dimension n, v is harmonic in D, and u \leq v, then v is called a harmonic majorant of u. If a harmonic majorant exists, then there exists the least harmonic majorant, and u(x) = v(x) - \int_D\frac,\quad n\geq 3 while in dimension 2, u(x) = v(x) + \int_D\log, x-y, d\mu(y), where v is the least harmonic majorant, and \mu is a
Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...
in D. This is called the Riesz representation theorem.


Subharmonic functions in the complex plane

Subharmonic functions are of a particular importance in
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, where they are intimately connected to
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
s. 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 \Complex 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 \int_0^ \varphi(z+ re^) \, d\theta. Intuitively, this means that a subharmonic function is at any point no greater than the
average In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...
of the values in a circle around that point, a fact which can be used to derive the
maximum principle In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. ...
. If f is a holomorphic function, then \varphi(z) = \log \left, f(z) \ is a subharmonic function if we define the value of \varphi(z) at the zeros of f to be −∞. It follows that \psi_\alpha(z) = \left, f(z) \^\alpha is subharmonic for every ''α'' > 0. This observation plays a role in the theory of Hardy spaces, especially for the study of ''H'' when 0 < ''p'' < 1. In the context of the complex plane, the connection to the
convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poi ...
s can be realized as well by the fact that a subharmonic function f on a domain G \subset \Complex that is constant in the imaginary direction is convex in the real direction and vice versa.


Harmonic majorants of subharmonic functions

If u is subharmonic in a
region In geography, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics ( physical geography), human impact characteristics ( human geography), and the interaction of humanity an ...
\Omega of the complex plane, and h is
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...
on \Omega, then h is a harmonic majorant of u in \Omega if u \leq h in \Omega. Such an inequality can be viewed as a growth condition on u.


Subharmonic functions in the unit disc. Radial maximal function

Let ''φ'' be subharmonic, continuous and non-negative in an open subset Ω of the complex plane containing the closed unit disc ''D''(0, 1). The ''radial maximal function'' for the function ''φ'' (restricted to the unit disc) is defined on the unit circle by (M \varphi)(e^) = \sup_ \varphi(re^). If ''P''''r'' denotes the Poisson kernel, it follows from the subharmonicity that 0 \le \varphi(re^) \le \frac \int_0^ P_r\left(\theta- t\right) \varphi\left(e^\right) \, dt, \ \ \ r < 1. It can be shown that the last integral is less than the value at ''e'' of the
Hardy–Littlewood maximal function In mathematics, the Hardy–Littlewood maximal operator ''M'' is a significant non-linear operator used in real analysis and harmonic analysis. Definition The operator takes a locally integrable function ''f'' : R''d'' → C and returns another ...
''φ'' of the restriction of ''φ'' to the unit circle T, \varphi^*(e^) = \sup_ \frac \int_^ \varphi\left(e^\right) \, dt, so that 0 ≤ ''M'' ''φ'' ≤ ''φ''. It is known that the Hardy–Littlewood operator is bounded on ''L''''p''(T) when 1 < ''p'' < ∞. It follows that for some universal constant ''C'', \, M \varphi\, _^2 \le C^2 \, \int_0^ \varphi(e^)^2 \, d\theta. If ''f'' is a function holomorphic in Ω and 0 < ''p'' < ∞, then the preceding inequality applies to ''φ'' = , ''f'', . It can be deduced from these facts that any function ''F'' in the classical Hardy space ''Hp'' satisfies \int_0^ \left( \sup_ \left, F(r e^)\ \right)^p \, d\theta \le C^2 \, \sup_ \int_0^ \left, F(re^)\^p \, d\theta. With more work, it can be shown that ''F'' has radial limits ''F''(''e'') almost everywhere on the unit circle, and (by the dominated convergence theorem) that ''Fr'', defined by ''Fr''(''e'') = ''F''(''r'e'') tends to ''F'' in ''L''''p''(T).


Subharmonic functions on Riemannian manifolds

Subharmonic functions can be defined on an arbitrary
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
. ''Definition:'' Let ''M'' be a Riemannian manifold, and f:\; M \to \R an upper semicontinuous function. Assume that for any open subset U\subset M, and any harmonic function ''f''1 on ''U'', such that f_1 \geq f on the boundary of ''U'', the inequality f_1 \geq 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 f \geq 0, where \Delta is the usual
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
.,


See also

* Plurisubharmonic function — generalization to
several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...
* Classical fine topology


Notes


References

* * * * {{Authority control Potential theory Complex analysis Types of functions