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A metric approach to elastic reformations. (English) Zbl 1316.49051

Summary: We study a variational framework to compare shapes, modeled as Radon measures on \(\mathbb R^N\), in order to quantify how they differ from isometric copies. To this purpose we discuss some notions of weak deformations termed reformations as well as integral functionals having some kind of isometries as minimizers. The approach pursued is based on the notion of pointwise Lipschitz constant leading to a metric space framework. In particular, to compare general shapes, we study this reformation problem by using the notion of transport plan and Wasserstein distances as in optimal mass transportation theory.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
49Q20 Variational problems in a geometric measure-theoretic setting
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
37J50 Action-minimizing orbits and measures (MSC2010)
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