Calculus of Variations and Geometric Measure Theory
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G. Canevari - A. Segatti - M. Veneroni

Morse's index formula in VMO for compact manifolds with boundary

created by canevari on 29 Jan 2021

[BibTeX]

preprint

Inserted: 29 jan 2021

Year: 2014

ArXiv: 1407.1707 PDF

Abstract:

In this paper, we study Vanishing Mean Oscillation vector fields on a compact manifold with boundary. Inspired by the work of Brezis and Niremberg, we construct a topological invariant - the index - for such fields, and establish the analogue of Morse's formula. As a consequence, we characterize the set of boundary data which can be extended to nowhere vanishing VMO vector fields. Finally, we show briefly how these ideas can be applied to (unoriented) line fields with VMO regularity, thus providing a reasonable framework for modelling a surface coated with a thin film of nematic liquid crystals.

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