Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 497-521

L¹-ELLIPTIC REGULARITY AND H = W ON THE WHOLE L^p-SCALE ON ARBITRARY MANIFOLDS

Davide Guidetti, Batu Güneysu and Diego Pallara

Università di Bologna, Dipartimento di Matematica
Piazza di Porta San Donato 5, 40126 Bologna, Italy; davide.guidetti 'at' unibo.it

Humboldt-Universität zu Berlin, Institut für Mathematik
10099 Berlin, Germany; gueneysu 'at' math.hu-berlin.de

Università del Salento, Dipartimento di Matematica e Fisica "Ennio De Giorgi" and I.N.F.N.
Piazza Tancredi, n7, 73100 Lecce, Italy; diego.pallara 'at' unisalento.it

Abstract. We define abstract Sobolev type spaces on L^p-scales, p ∈ [1,∞), on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families P of linear partial differential operators, and we prove the density of the corresponding smooth Sobolev sections in these spaces under a generalised ellipticity condition on the underlying family. In particular, this implies a covariant version of Meyers–Serrin's theorem on the whole L^p-scale, for arbitrary Riemannian manifolds. Furthermore, we prove a new local elliptic regularity result in L¹ on the Besov scale, which shows that the above generalised ellipticity condition is satisfied on the whole L^p-scale, if some differential operator from P that has a sufficiently high (but not necessarily the highest) order is elliptic.

2010 Mathematics Subject Classification: Primary 35J45, 46E35, 58JXX.

Key words: Meyers–Serrin Theorem, L¹-regularity theory for elliptic systems, Sobolev spaces on manifolds.

Reference to this article: D. Guidetti, B. Güneysu and D. Pallara: L¹-elliptic regularity and H = W on the whole L^p-scale on arbitrary manifolds. Ann. Acad. Sci. Fenn. Math. 42 (2017), 497-521.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4234

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