On the Applicability of the Fredholm-Radon Method in Potential Theory and the Panel Method

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

“…Then u is a weak bounded solution of the third problem for the Laplace equation with the boundary condition µ. If c < 1 and (G) of (14). If there is µ ∈ C (G)…”

“…(G) of the third problem (14). If u has the form (10), where c j = 0 for G j unbounded, then u is a weak solution of this third problem.…”

“…(G) of the third problem for the Laplace equation (14) then u is bounded in G if and only if µ ∈ C b (∂G). [17], Theorem 1 yields that µ ∈ C 0 (∂G).…”

Solution of the Dirichlet Problem for the Laplace equation

“…Since V B − βI is a Fredholm operator with index 0 by Lemma 5, [17], Satz 51.8, Satz 51.1, the spectral radius of the operator V B − αI is smaller than |α|. Therefore there are constants d α ∈ 1, ∞), q α ∈ (0, 1) such that the relation (22) holds for each g ∈ B(∂G) and an integer number n. Since V is invertible in B there is a unique f ∈ B(∂G) such that V f = g for each g ∈ B(∂G). Easy calculation yields that f given by the series (23) fulfils V f = g.…”

The third boundary value problem in potential theory for domains with a piecewise smooth boundary

“…For each v £ #" we define the distribution 2fv by Assume G to be a domain with do(y) ^ 0 for every y € B and suppose that (5) inf -^ < 1. where n is a positive integer, vrij G #" and /js are bounded Baire functions on B. However in [33] an example is given of a rectangular domain G in IR 3 such that the condition (5) is not fulfilled even for A = 0. We shall substitute the condition (5) by a weaker condition and then we shall prove the result of [46].…”