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. 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. 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.