2.1 An Iterative Process for Solving the Problem of Equilibrium Charge Distribution and Charge Distribution on a Conductor Placed in an Exterior Static Field

1. In Section 1.3, Problem 1.1 from Section 1.1 was reduced to problem (1.32). It is known [38] that the operator A in (1.32) is compact in L ²( ?) and in C( ?) provided that ? is smooth (it is sufficient to assume that the equation of the surface in the local coordinates is x ₃ = f( x ₁ , x ₂) and ? f is H lder continuous). It is also known [38] that ? = -1 is the smallest characteristic value of A which is simple. This means that ? = -1 is a simple pole of the resolvent ( A - ?I) ^-1 and the corresponding null space is one-dimensional, i.e., every solution of the equation ? = - A ? is of the form ? = const ?( t), where ?( t) is the solution normalized by the condition ? _? ? dt = 1. Let G ₁ denote the null space of the operator I + A*, where A* is defined in (1.28). It is known [38] and can be verified directly that ? = 1 is a solution of the equation ? = - A* ?.