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 gravi ...
.
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 a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...
s of one variable as follows. If the
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
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
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
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
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, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
and let
be an
upper semi-continuous function. Then,
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 defin ...
of center
and radius
contained in
and every
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
-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 value ...
on
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'', the ...
in
and satisfies
for all
on the
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
of
, we have
for all
Note that by the above, the function which is identically −∞ is subharmonic, but some authors exclude this function by definition.
A function
is called ''superharmonic'' if
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'', the ...
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 bicondi ...
it is both subharmonic and superharmonic.
* If
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 suf ...
in
, then
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 bicondi ...
one has
on
, where
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 ran ...
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 ran ...
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 .
...
, that is, a linear combination of subharmonic functions with positive coefficients is also subharmonic.
*The
pointwise maximum
In mathematics, the lower envelope or pointwise minimum of a finite set of functions is the pointwise minimum of the functions, the function whose value at every point is the minimum of the values of the functions in the given set. The concept of ...
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
).
*Subharmonic functions are not necessarily continuous in the usual topology, however one can introduce the
fine topology which makes them continuous.
Examples
If
is
analytic then
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
is subharmonic in a region
, in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
of dimension
,
is harmonic in
, and
, then
is called a harmonic majorant of
. If a harmonic majorant exists, then there exists the least harmonic majorant, and
while in dimension 2,
where
is the least harmonic majorant, and
is a
Borel measure in
.
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 derivativ ...
s.
One can show that a real-valued, continuous function
of a complex variable (that is, of two real variables) defined on a set
is subharmonic if and only if for any closed disc
of center
and radius
one has
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
is a holomorphic function, then
is a subharmonic function if we define the value of
at the zeros of
to be −∞. It follows that
is subharmonic for every ''α'' > 0. This observation plays a role in the theory of
Hardy spaces
In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . ...
, 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 a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...
s can be realized as well by the fact that a subharmonic function
on a domain
that is constant in the imaginary direction is convex in the real direction and vice versa.
Harmonic majorants of subharmonic functions
If
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 and t ...
of the complex plane, and
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'', the ...
on
, then
is a harmonic majorant of
in
if
in
. Such an inequality can be viewed as a growth condition on
.
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
If ''P''
''r'' denotes the
Poisson kernel
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the deriv ...
, it follows from the subharmonicity that
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,
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'',
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 ''H
p'' satisfies
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
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary ...
) that ''F
r'', defined by ''F
r''(''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 manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
.
''Definition:'' Let ''M'' be a Riemannian manifold, and
an
upper semicontinuous
In mathematical analysis, semicontinuity (or semi-continuity) is a property of Extended real number, extended real-valued Function (mathematics), functions that is weaker than Continuous function, continuity. An extended real-valued function f is ...
function. Assume that for any open subset
, and any
harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
''f''
1 on ''U'', such that
on the boundary of ''U'', the inequality
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
, where
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 In mathematics, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subset of the subharmonic f ...
— 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