A boundary value problem with a conormal derivative and a nonlocal integral condition for a degenerate elliptic equation of the second kind

Abstract

A boundary value problem for a degenerate elliptic equation of the second kind in a finite domain bounded by a smooth curve and a segment of the degeneracy line is investigated. A condition with a conormal derivative, generalizing the Poincaré condition, is specified on the curve, and a nonlocal integral condition containing a normal derivative and negative-order fractional Riemann-Liouville integrals is specified on the degeneracy line. The uniqueness of the solution is established using the energy integral method under the condition that the product of the boundary condition coefficients is nonnegative and the nonlocal condition coefficients have a definite sign. The existence of a solution is proved for a nonzero coefficient of the conormal derivative by reducing to an auxiliary problem with a given trace on the degeneracy line: a Green's function is constructed, and a functional relationship is derived between the trace of the solution and its normal derivative, which, using lemmas on the transformation of fractional integrals, is reduced to a singular integral equation of normal type with a zero index. Using the Carleman-Vekua regularization method, it is reduced to a Fredholm equation of the second kind, the solvability of which follows from the uniqueness of the solution. The results develop the theory of boundary value problems with a conormal derivative for degenerate elliptic equations.