Harmonic Functions

functions of n variables (n≤2) that are continuous over some domain together with their partial derivatives of the first and second orders and that satisfy Laplace’s differential equation in this domain:

In many problems of physics and mechanics concerned with the state of a portion of space, which depends on the position of a point but not on time (equilibrium, steady-state motion, and so on), the corresponding state is represented by a harmonic function of the point’s coordinates. For example, the potential of the gravitational forces in a region that contains no attracting masses and the potential of a constant electrical field in a region containing no electrical charges are harmonic functions. The potential of the velocities of a steady-state irrotational motion of an incompressible fluid, the temperature of a body under conditions of a steady-state heat distribution, and the bending of a membrane stretched over an arbitrary contour that is completely nonplanar (the membrane’s weight is ignored) are also harmonic functions.

Harmonic functions of three variables (the coordinates of a point) are the most important for use in physics and mechanics. In a particular case, when the region of space is bounded by a cylindrical surface whose generatrices are parallel—for example, the z axes—the phenomenon under investigation proceeds in the same manner in any plane perpendicular to the generatrices (it does not depend on the z coordinate), the corresponding harmonic functions of three variables are converted into harmonic functions of two variables, x and y. The latter are closely related to the analytic functions f(ξ) of a complex variable ξ = x + iy. Each harmonic function of x and y is a real or imaginary part of some function f(ξ), and conversely, the real and imaginary parts of any analytic function are harmonic functions of x and y. For example, x² − y² and 2 xy, being the real and imaginary parts of the function ξ² = x² − y² + 2 ixy, are harmonic functions. The major problems of the theory of harmonic functions are boundary-value problems, in which it is necessary to find a harmonic function within a region using data concerning the behavior of the function on the boundary of the region. An example is the Dirichlet problem, in which a harmonic function is sought from its values on the region’s boundary points (for example, the determination of the temperature within a body from its surface temperature, which is maintained in such a way that it depends on position but not on time, or the determination of the form of a membrane from the form of the contour over which it is stretched). There is also the Neumann problem, in which a harmonic function is sought from the value of its normal derivative, which is specified on the boundary of a region (for example, the determination of the temperature within a body from the temperature gradient on its surface, or the determination of the potential of the motion of an incompressible fluid around a solid object, based on the fact that the normal components of the velocities of the particles of fluid that adjoin the object’s surface coincide with the specified normal components of the velocities of points on the surface).

Various methods of great theoretical value have been developed to solve the Dirichlet, Neumann, and other boundary-value problems of the theory of harmonic functions. Known methods include, for the Dirichlet problem, the alternating method (Schwartz’s method), the sweeping-out method (Poincaré’s method), the method of integral equations (Fredholm’s method), and the method of upper and lower functions (Perron’s method). In considering boundary-value problems for regions of a general form, important problems arise concerning the conditions of existence of solutions and the stability of solutions for small variations of the region’s boundary. These problems are the topic of the work of M. V. Keldysh, M. A. Lavrent’ev, and other contemporary Soviet mathematicians. Numerical methods of solving boundary-value problems have been developed for use in physics and technology.