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

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

“…by [16], Lemma 1.5, the operator W +V is Fredholm. Since σ, U B µ = 0, we conclude that U B µ ∈ (W + V )(B) by [29], Chapter VII, Theorem 3.1.…”

“…Since τ < α by [17], Lemma 2, there is no eigenvalue β = 0 of τ such that |α − β| α. According to [16], Lemma 1.2, Lemma 1.5 we have r ess (W + V − αI) = r ess (W + V − αI) = r ess (τ − αI) < α. If β is an eigenvalue of W + V then β is an eigenvalue of τ , because W + V is the restriction of τ to B.…”

“…In [16] it was shown that the condition r ess (τ − 1 2 I) < 1 2 has a local character. As a conclusion we obtain that this condition is fullfiled for G ⊂ 3 such that for each x ∈ ∂G there are r(x) > 0, a domain D x which is polyhedral or smooth or convex or a complement of a convex domain and a diffeomorphism ψ x : U (x; r(x)) → In the rest of the paper we will suppose that r ess (τ −…”

“…there exists a continuous function U c σ on m coinciding with U σ on m \ ∂G (see [16], Theorem 1.11, [17], Lemma 13). According to [19], Lemma 3 the set G has finitely many components…”

“…(Here χ ∂Gj denotes the characteristic function of ∂G j .) According to [16], Lemma 1.5 the operator W G is a bounded…”