Solution of the Robin problem for the Laplace equation

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

“…VII, § § 2, 6, [15], Theorem 5.11, Theorem 5.10) and U µ = U µ + − U µ − is finite and fine-continuous outside of a polar set. [17], Remark 6).…”

“…compact operator (see [17], Remark 5), this condition is equivalent to the condition [18], Lemma 2). If x ∈ ∂G then d G (x) exists and is strictly positive (see [17], Lemma 14).…”

“…It was shown in [17] that U ν has all the properties of the solution of the third problem with some boundary condition, but our "continuity" on the boundary is replaced by the fine continuity at λ-a.a. points of the boundary. If U ν is fine-continuous in x ∈ ∂G with respect to cl G then u(x) is the approximative limit of u at x over G (see [11], Theorem 10.15, Corollary 10.5).…”

“…If U ν is fine-continuous in x ∈ ∂G with respect to cl G then u(x) is the approximative limit of u at x over G (see [11], Theorem 10.15, Corollary 10.5). If U ν is a solution of the third problem in the sense of [17] then it is a weak solution of the third problem.…”

“…by [17], Corollary 1 and therefore m \ ∂G is finely dense in m (see [2], Chap. VII, § § 2, 6, [15], Theorem 5.11, Theorem 5.10) and U µ = U µ + − U µ − is finite and fine-continuous outside of a polar set.…”