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Curvature-dependent energies: a geometric and analytical approach

Part of: Real functions Existence theories Manifolds

Published online by Cambridge University Press:  27 February 2017

Emilio Acerbi  and
Domenico Mucci
Emilio Acerbi
Affiliation:
Dipartimento di Matematica e Informatica, Università di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy ([email protected]; [email protected])
Domenico Mucci
Affiliation:
Dipartimento di Matematica e Informatica, Università di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy ([email protected]; [email protected])
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Extract

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

Keywords

curvestotal curvaturerelaxationbounded variationscurrentsimage restoration

MSC classification

Secondary: 49J45: Methods involving semicontinuity and convergence; relaxation 49Q15: Geometric measure and integration theory, integral and normal currents 26A45: Functions of bounded variation, generalizations 65K10: Optimization and variational techniques
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 147 , Issue 3 , June 2017 , pp. 449 - 503
Copyright
Copyright © Royal Society of Edinburgh 2017 

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