Oblique Derivative Problem

Abstract

Having proved in Chapter 9 that the Dirichlet problem on a spherical chip subject to a normal derivative condition on the flat portion of the boundary and a Dirichlet type condition on the remaining portion has a solution, it is shown that such a solution can be morphed onto a local solution of an elliptic equation on a neighborhood of a boundary point satisfying mixed boundary conditions. The Perron-Wiener-Brelot method is then used to prove the existence of a global solution on a bounded open subset of $R^n$ that satisfies an oblique derivative boundary condition on a relatively open subset of the boundary and Dirichlet type condition on the remainder of the boundary. The usual criteria for regularity at points of the latter part of the boundary apply but the border points of the two types of subsets require a more restrictive “wedge condition” rather than a Zaremba cone condition. The chapter concludes with a theorem that establishes that the range of the resolvent of the elliptic operator is sufficient for applying the Hille-Yosida Theorem in the next chapter.