Calculus of Variations and Geometric Measure Theory

E. Stepanov - D. Trevisan

Towards geometric integration of rough differential forms

created by trevisan on 22 Nov 2017
modified by stepanov on 02 Apr 2020


Published Paper

Inserted: 22 nov 2017
Last Updated: 2 apr 2020

Journal: J. Geom. Anal.
Year: 2020

ArXiv: 2001.06446 PDF


We provide a draft of a theory of geometric integration of ``rough differential forms'' which are generalizations of classical (smooth) differential forms to similar objects with very low regularity, for instance, involving Hölder continuous functions that may be nowhere differentiable. Borrowing ideas from the theory of rough paths, we show that such a geometric integration can be constructed substituting appropriately differentials with more general asymptotic expansions. This can be seen as the basis of geometric integration similar to that used in geometric measure theory, but without any underlying differentiable structure, thus allowing Lipschitz functions and rectifiable sets to be substituted by far less regular objects (e.g.\ Hölder functions and their images which may be purely unrectifiable). Our construction includes both the one-dimensional Young integral and multidimensional integrals introduced recently by R. Züst. To simplify the exposition, we limit ourselves to integration of forms of dimensions not exceeding two.

Keywords: Exterior differential calculus, Young integral, Stokes Theorem, Rough paths, Sewing lemma, Hölder functions