Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2019
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/1905.02661 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
Sommario:
- We are concerned with the global weak continuity of the Cartan structural system -- or equivalently, the Gauss--Codazzi--Ricci system -- on semi-Riemannian manifolds with lower regularity. For this purpose, we first formulate and prove a geometric compensated compactness theorem on vector bundles over semi-Riemannian manifolds with lower regularity (Theorem 3.2), extending the classical quadratic theorem of compensated compactness. We then deduce the $L^p$ weak continuity of the Cartan structural system for $p>2$: For a family $\{\mathcal{W}_\varepsilon\}$ of connection $1$-forms on a semi-Riemannian manifold $(M,g)$, if $\{\mathcal{W}_\varepsilon\}$ is uniformly bounded in $L^p$ and satisfies the Cartan structural system, then any weak $L^p$ limit of $\{\mathcal{W}_\varepsilon\}$ is also a solution of the Cartan structural system. Moreover, it is proved that isometric immersions of semi-Riemannian manifolds into semi-Euclidean spaces can be constructed from the weak solutions of the Cartan structural system or the Gauss--Codazzi--Ricci system (Theorem 5.1), which leads to the $L^p$ weak continuity of the Gauss--Codazzi--Ricci system on semi-Riemannian manifolds. As further applications, the weak continuity of Einstein's constraint equations, general immersed hypersurfaces, and the quasilinear wave equations is also established.