Application of Scans and Fractional Power Integrands
In this note we describe the notion of a rectifiable scan and consider some applications [DH1], [DH2] to Plateau-type minimization problems. “Scans” were first introduced in the work [HR1] of Tristan Rivière and the second author to adequately describe certain bubbling phenomena. There, the behaviour of certain W 1,3 weakly convergent sequences of smooth maps from four-dimensional domains into S 2 led to the consideration of a necessarily infinite mass generalization of a rectifiable current. The definition of a scan is motivated by the fact that a rectifiable current can be expressed in terms of its lower-dimensional slices by oriented affine subspaces. By integral geometry, the slicing function for the rectifiable current is a mass integrable function of the subspaces. With a scan one considers more general such functions that are not necessarily mass integrable.
KeywordsSoap Film Geometric Measure Theory Rectifiable Current Interior Regularity Plateau Problem
Unable to display preview. Download preview PDF.
- [A2]F.J.Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Amer. Math. Soc. 4(1976), no. 165.Google Scholar
- [D]E. De Giorgi, Frontiere orientate di misura minima. Sem. Mat. Scuola Norm. Sup. Pisa, 1961.Google Scholar
- [DH2]T. De Pauw and R. Hardt, Partial regularity of scans minimizing a fractional density mass. In preparation.Google Scholar
- [DH3]T. De Pauw and R. Hardt, Rectifiable scans in a metric space. In preparation.Google Scholar
- [HR1]R. Hardt and T. Rivière, Connecting topological Hopf singularities. To appear in Annali Sc. Norm. Sup. Pisa.Google Scholar
- [HR2]R. Hardt and T. Rivière, Connecting rational homotopy type singularities. In preparation.Google Scholar
- [S]L. Simon, Lectures on geometric measure theory. Proc. Centre for Math. Anal. 3 (1983) Australian National University, Canberra.Google Scholar
- [X2]Q. Xia, Interior regularity of optimal transport paths. Preprint 2002.Google Scholar