The solution for mixed boundary value problems of two-dimensional potential theory (Q1917710)

The authors consider and solve two boundary problems for a harmonic, bidimensional function in the infinite strip \(S=\{(x,y):-\infty\leq x\leq\infty,\;0\leq y\leq 1\}\) of the complex plane \(\mathbb{C}\). The first problem: find the harmonic function \(V(x,y)\) in \(S\) with the boundary conditions enumerated below; a) \(\partial V/\partial y=0\) for \(|x|\geq 1\), \(y=0\); b) \(V(x,y)=g(x)\) for \(|x|\leq 1\), \(y=0\); c) \(V(x,y)=0\) for \(y=L/a\); d) \(|V(x,y)|<\infty\) for \(|x|\to\infty\). The second problem: find the harmonic function \(V(x,y)\) in \(S\) with the same boundary conditions where c) is replaced by c') \(\partial V/\partial y=0\) for \(y=L/a\). The solutions are given by means of Green's functions. The above problems have applications in electricity.