The Fredholm Method in Potential Theory

Solution of the Dirichlet Problem for the Laplace equation

“…It was shown by J. Král (see [21]) and independently by Yu. D. Burago and V. G. Maz'ya (see [7]) that it is possible to define the double layer potential on G as a continuously extendable function on cl G for each density f ∈ C (∂G) if and only if the cyclic variation of G is bounded, where If z ∈ Ê m and θ is a unit vector such that the symmetric difference of G and the half-space {x ∈ Ê m ; (x − z) · θ > 0} has m-dimensional density zero at z then n G (z) = θ is termed the interior normal of G at z in Federer's sense.…”

“…is fulfilled for sets with a smooth boundary (of class C 1+α ) (see [21]) and for convex sets (see [31] [16]). (Let us note that there is a polyhedral set in Ê 3 which has not a locally Lipschitz boundary (see [27], Example 2).)…”

Boundedness of the solution of the third problem for the Laplace equation

“…It is well-known that the condition r ess τ − 1 2 I < 1 2 is fulfilled for sets with a smooth boundary (of class C 1+α ) (see [13]) and for convex sets (see [23] 2 ) and ∂Ω is formed by a finite number of plane angles.…”

“…Since we can take arbitrary ν we According to [24], Proposition 8, [13] we may define on B continuous operators V ,…”

A numerical solution of the dirichlet problem on some special doubly connected regions

“…Note that [8] deals with non-tangential limits. For the general case of Ê n , n 2, see [12], [10]. It is easy to show that the set {ζ ∈ K | |∆(ζ)| > 0} formed by the angular points of K is at most countable (cf.…”