A New Proof of Gaffney’s Inequality for Differential Forms on Manifolds-with-Boundary: The Variational Approach à La Kozono-Yanagisawa

Abstract

Let (\({\cal M}\), g0) be a compact Riemannian manifold-with-boundary. We present a new proof of the classical Gaffney inequality for differential forms in boundary value spaces over \({\cal M}\), via a variational approach à la Kozono-Yanagisawa [Lr-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains, Indiana Univ. Math. J. 58 (2009), 1853–1920], combined with global computations based on the Bochner technique.