Weak limits of quasiminimizing sequences. (English) Zbl 1482.49013
This paper is motivated by classical variational Plateau’s problem relying on area minimization of the surface spanning a boundary. In practice, physical experiments produce nice looking soap film shapes of minimal area such as created e.g. with the help of Surface Evolver program by K. Brakke (http://facstaff.susqu.edu/brakke/evolver/evolver.html). From theoretical point of view the problem is a difficult one. Therefore, in literature there are many approaches (see: [G. David, Princeton Math. Ser. 50, 108–145 (2014; Zbl 1304.49083)]) to solving such problems depending on the used mathematical model and classes of admissible surfaces (called as competitors) in which the best solutions are found.
The author is interested in models that allow to describe the singularities of soap films. He considers the problem with compact boundary \(\Gamma\) in \(\mathbb R^n\) and the goal is to minimize Hausdorff measure \(\mathcal{H}^d, d = 1,\ldots,n\) of competitors.
The main result of the paper says that the weak limit of a quasiminimizing sequence is a quasiminimal set. It is analogous result to the limiting theorem of [G. David, Local regularity properties of almost- and quasiminimal sets with a sliding boundary condition. Paris: Société Mathématique de France (SMF) (2019; Zbl 1428.49001)] for sequences of quasiminimal sets which converge in local Hausdorff distance. That result also generalizes the solution strategy presented in: (see: [C. De Lellis et al., J. Eur. Math. Soc. (JEMS) 19, No. 8, 2219–2240 (2017; Zbl 1376.49051)] and [G. De Philippis et al., Adv. Math. 288, 59–80 (2016; Zbl 1335.49067)]). The author shows that in his approach it is also possible to minimize the intersection of competitors with the boundary. In the proof suitable sliding deformations are constructed.
A large part of the paper is devoted to description of a structure for building Federer-Fleming projections needed in construction of competitors and the proof of the direct solution method.
The paper is a long one (113 pages, 812 equations), but fortunately it is self-contained and contains all needed facts to follow proofs. The bibliography containing 25 items creates a good background for understanding the paper.
The author is interested in models that allow to describe the singularities of soap films. He considers the problem with compact boundary \(\Gamma\) in \(\mathbb R^n\) and the goal is to minimize Hausdorff measure \(\mathcal{H}^d, d = 1,\ldots,n\) of competitors.
The main result of the paper says that the weak limit of a quasiminimizing sequence is a quasiminimal set. It is analogous result to the limiting theorem of [G. David, Local regularity properties of almost- and quasiminimal sets with a sliding boundary condition. Paris: Société Mathématique de France (SMF) (2019; Zbl 1428.49001)] for sequences of quasiminimal sets which converge in local Hausdorff distance. That result also generalizes the solution strategy presented in: (see: [C. De Lellis et al., J. Eur. Math. Soc. (JEMS) 19, No. 8, 2219–2240 (2017; Zbl 1376.49051)] and [G. De Philippis et al., Adv. Math. 288, 59–80 (2016; Zbl 1335.49067)]). The author shows that in his approach it is also possible to minimize the intersection of competitors with the boundary. In the proof suitable sliding deformations are constructed.
A large part of the paper is devoted to description of a structure for building Federer-Fleming projections needed in construction of competitors and the proof of the direct solution method.
The paper is a long one (113 pages, 812 equations), but fortunately it is self-contained and contains all needed facts to follow proofs. The bibliography containing 25 items creates a good background for understanding the paper.
Reviewer: Wiesław Kotarski (Sosnowiec)
MSC:
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 49Q05 | Minimal surfaces and optimization |
| 49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |
Keywords:
Plateau problem; soap films; Hausdorff measure; weak convergence; Reifenberg competitor; Federer-Fleming projections; sliding deformation; (quasi)minimal set; admissible energy; rectifiabilityReferences:
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