The LIR Method. $L^r$ solutions of elliptic equation in a complete riemannian manifold

Abstract

We introduce the Local Increasing Regularity Method (LIRM) which allows us to get from \emph{local} a priori estimates, on solutions $u$ of a linear equation $Du=\omega ,$ \emph{global} ones. As an application we shall prove that if $D$ is an elliptic linear differential operator of order $m$ with ${\mathcal{C}}^{\infty }$ coefficients operating on the sections of a complex vector bundle $G:=(H,\pi ,M)$ over a compact Riemannian manifold $M$ without boundary and $\omega \in L_G^r(M)\cap (\ker D^{\ast }{)}^{\perp },$ then there is a $u\in W_G^{m,r}(M)$ such that $Du=\omega$ on $M.$ \quad Next we investigate the case of a compact manifold with boundary by use of the "riemannian double manifold". In the last sections we study the more delicate case of a complete but non compact Riemannian manifold by use of adapted weights.

arXiv commentThis version generalises the previous results to the case of vector bundles on complete riemannian manifold. Some small corrections are also done