In mathematics, subharmonic and superharmonicfunctions 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
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.
Properties
A function is harmonic if and only if it is both subharmonic and superharmonic.
If is C2 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.
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.
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 plane
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 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 ofHardy 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 functions
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.
Let φ be subharmonic, continuous and non-negative in an open subsetΩ of the complex plane containing the closed unit discD. The radial maximal function for the function φ 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 functionφ∗ 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 when 1 < p < ∞. It follows that for some universal constantC, 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 Falmost everywhere on the unit circle, and that Fr, defined by Fr = F tends to F in Lp.
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.
