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In this article, we discuss that the discontinuous oblique derivative boundary value problem for nonlinear uniformly elliptic system of second order equations in multiply connected domains. We first propose the discontinuous oblique derivative problem and its new modified well-posedness. Next we give a priori estimates of solutions of the modified discontinuous boundary value problem for corresponding elliptic system of first order complex equations and verify its solvability by the above estimates of solutions and the Leray-Schauder theorem. Finally the solvability results of the original discontinuous oblique derivative problem can be derived. Here we mention the discontinuous boundary value problems possess many applications in mechanics and physics etc.

DISCONTINUOUS OBLIQUE DERIVATIVE PROBLEM FOR NONLINEAR ELLIPTIC SYSTEM OF SECOND ORDER EQUATIONS IN MULTIPLY CONNECTED DOMAINS