A varifold perspective on the \(p\)-elastic energy of planar sets. (English) Zbl 1446.49032
The elastic properties of the boundaries of measurable sets in the plane are studied. A new definition of the sets which are enough regular for having finite \(p\)-elastic energy is given. A basic inequality concerning the elastic energy of immersed curves is proved using a varifold perspective. It is applied to prove the structural properties of elastic varifolds. Then, \(L^1\)-relaxation is characterized for the energy \(F_p\), \(p>1\), extending the definition of regular sets by an intrinsic varifold perspective. This relaxation is compared with the classical one of G. Bellettini and L. Mugnai. Finally, an application to the inpainting problem is discussed and some examples are given. The paper is complete and clearly written.
Reviewer: Stepan Agop Tersian (Rousse)
MSC:
| 49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |
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