General Discontinuous Irregular Oblique Derivative Problems for Nonlinear Elliptic Equations of Second Order

It is known that in mechanics and physics, many problems have discontinuous boundary conditions (see [1]-[4]). In this paper, the discontinuous irregular oblique derivative problem (i.e. the discontinuous Poincar´e boundary value problem) for nonlinear elliptic equations of second order in multiply connected domains is discussed. We first prove the uniqueness of solutions for the above boundary value problem, and then give a priori estimates of its solutions. Moreover by the Leray-Schauder theorem and compactness principle, the existence of solutions of the above problem for the second order equations is proved.