The Laplace and Poisson Equations for a Normal Domain
Efim M. Polishchuk · Birkhäuser Basel eBooks · 1988
In this chapter we consider Dirichlet’s problem in a function space in the following setting. Let V be a normal domain in the space L 2 (q) bounded by the surface S. Find a functional H[X] satisfying, for each sphere Ω a, R ⊂ V, the condition m a, R H = H [a] and taking on S the given value F[x], where F is an element of the Gâteaux ring. The proofs are based on semigroup properties of the mean m a, R . It is shown that the solution can be represented in a closed form if the “fundamental functional of the surface S ” is known. This functional can be effectively found for surfaces which are called algebraic as well as in more general cases. Then the same idea is applied to the boundary value problem for Poisson’s equation. The exterior Dirichlet problem is also examined. A complex domain is needed to cope with this problem, and additional conditions have to be imposed on the prescribed functions. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.