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

be a subset of the Euclidean space be an upper semi-continuous function is called subharmonic if for any closed ball and every real-valued continuous function that is harmonic on the boundary

Note that by the above, the function which is identically −∞ is subharmonic, but some authors exclude this function by definition.

is called superharmonic is subharmonic.

Properties

• A function is harmonic if and only if it is both subharmonic and superharmonic.

• is C2 (twice continuously differentiable) on an open set is subharmonic if and only if is the Laplacian.

• The maximum of a subharmonic function cannot be achieved in the interior of its domain unless the function is constant, which is called the 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. If the pointwise maximum of a countable number of subharmonic functions is upper semi-continuous, then it is also subharmonic.

• ).

• Subharmonic functions are not necessarily continuous in the usual topology, however one can introduce the fine topology which makes them continuous.

Examples

is analytic 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

, in Euclidean space is a Borel measure. This is called the Riesz representation theorem.

Subharmonic functions in the complex plane

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

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.

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 functions that is constant in the imaginary direction is convex in the real direction and vice versa.

Harmonic majorants of subharmonic functions

is subharmonic in a region is harmonic is a harmonic majorant.

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 If Pr denotes the Poisson kernel It can be shown that the last integral is less than the value at e of the Hardy–Littlewood maximal function φ∗ of the restriction of φ to the unit circle T so that 0≤ Mφ≤ 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|. It can be deduced from these facts that any function F in the classical Hardy space Hp 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(re) tends to F in Lp(T).

Subharmonic functions on Riemannian manifolds

Subharmonic functions can be defined on an arbitrary Riemannian manifold.

Definition: Let M an upper semicontinuous, and any harmonic function f1 on U on the boundary of U holds on all U. Then f is called subharmonic.

is the usual Laplacian.

See also

• Plurisubharmonic function — generalization to several complex variables

• Classical fine topology

References

Works cited

General references