$L^{r}$ solutions of elliptic equation in a complete riemannian manifold.
Langue
en
Article de revue
Ce document a été publié dans
Journal of Geometric Analysis. 2019, vol. 23, n° 3, p. 2565-2599
Résumé en anglais
Let $X$ be a complete metric space and $\Omega $ a domain in $ X.$ The Local Increasing Regularity Method (LIRM) allows to get from emph{local} a priori estimates on solutions $u$ of a linear equation $ Du=\omega $ ...Lire la suite >
Let $X$ be a complete metric space and $\Omega $ a domain in $ X.$ The Local Increasing Regularity Method (LIRM) allows to get from emph{local} a priori estimates on solutions $u$ of a linear equation $ Du=\omega $ \emph{global} ones in $ \Omega .$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 $p$-forms in a compact Riemannian manifold $M$ without boundary and $\omega \in L^{r}_{p}(M)\cap (\mathrm{k}\mathrm{e}mathrm{r}D^{*})^{\perp },$ then there is a $u\in W^{m,r}_{p}(M)$ such that $Du=\omega $ on $M.$ Next we investigate the case of a compact manifold with boundary. In the last sections we study the case of a complete but non compact Riemannian manifold by use of adapted weights.\< Réduire
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