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 definitionEdit

Formally, the definition can be stated as follows. Let   be a subset of the Euclidean space   and let

 

be an upper semi-continuous function. Then,   is called subharmonic if for any closed ball   of center   and radius   contained in   and every real-valued continuous function   on   that is harmonic in   and satisfies   for all   on the boundary   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.

PropertiesEdit

  • A function is harmonic if and only if it is both subharmonic and superharmonic.
  • If   is C2 (twice continuously differentiable) on an open set   in  , then   is subharmonic if and only if one has   on  , where   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. However, the minimum of a subharmonic function can be achieved in the interior of its domain.
  • Subharmonic functions make a convex cone, that is, a linear combination of subharmonic functions with positive coefficients is also subharmonic.
  • The pointwise maximum of two subharmonic functions is 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.

ExamplesEdit

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 TheoremEdit

If   is subharmonic in a region  , in Euclidean space 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 planeEdit

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   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 of the values in a circle around that point, a fact which can be used to derive the maximum principle.

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, especially for the study of Hp when 0 < p < 1.

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   on a domain   that is constant in the imaginary direction is convex in the real direction and vice versa.

Harmonic majorants of subharmonic functionsEdit

If   is subharmonic in a region   of the complex plane, and   is harmonic on  , then   is a harmonic majorant of   in   if    in  . Such an inequality can be viewed as a growth condition on  .[1]

Subharmonic functions in the unit disc. Radial maximal functionEdit

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 Pr denotes the Poisson kernel, it follows from the subharmonicity that

 

It can be shown that the last integral is less than the value at e iθ of the Hardy–Littlewood maximal function 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 Lp(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 | p/2. It can be deduced from these facts that any function F in the classical Hardy space Hp satisfies

 

With more work, it can be shown that F has radial limits F(e iθ) almost everywhere on the unit circle, and (by the dominated convergence theorem) that Fr, defined by Fr(e iθ) = F(r e iθ) tends to F in Lp(T).

Subharmonic functions on Riemannian manifoldsEdit

Subharmonic functions can be defined on an arbitrary Riemannian manifold.

Definition: Let M be a Riemannian manifold, and   an upper semicontinuous function. Assume that for any open subset  , and any harmonic function f1 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.[2]

See alsoEdit

NotesEdit

  1. ^ Rosenblum, Marvin; Rovnyak, James (1994), p.35 (see References)
  2. ^ Greene, R. E.; Wu, H. (1974). "Integrals of subharmonic functions on manifolds of nonnegative curvature". Inventiones Mathematicae. 27 (4): 265–298. doi:10.1007/BF01425500., MR0382723

ReferencesEdit

  • Rosenblum, Marvin; Rovnyak, James (1994). Topics in Hardy classes and univalent functions. Birkhauser Advanced Texts: Basel Textbooks. Basel: Birkhauser Verlag.

This article incorporates material from Subharmonic and superharmonic functions on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.