Elliptic Boundary Value Problems on a Manifold

In this chapter, following G. Schwarz (Hodge Decomposition – A Method for Solving Boundary Value Problems. Springer-Verlag, Berlin/Heidelberg/New York/Tokyo, 1995, Section 1.6 ) we establish the ellipticity of boundary value problems in the framework of vector bundles over an n-dimensional, compact smooth Riemannian manifold M with boundary \(\partial M\) . More precisely, we formulate the Shapiro–Lopatinskii ellipticity condition with a direct computational condition via the Fourier transform, and state the most fundamental theorem for the elliptic boundary value problem, the proof of which goes beyond the scope of this book.