Approximation of minimal surfaces with free boundaries. (English) Zbl 1448.53010
The authors develop a penalty method to approximate solutions of the free boundary problem for minimal surfaces. For this goal they study the problem of finding minimizers of a certain functional which is defined as the sum of the Dirichlet integral as well as an appropriate penalty term weighted by a parameter \(\lambda\). They show – among other things – the existence of a solution when \(\lambda\) is large enough and the convergence to a solution of the free boundary problem in the case when the parameter \(\lambda\) tends to infinity. Regularity at the boundary of these solutions is obtained.
Reviewer: Themistocles M. Rassias (Athína)
MSC:
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 49Q05 | Minimal surfaces and optimization |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
minimal surfaces; stationary surfaces; Dirichlet integral; harmonic solution; harmonic extension; free boundary problem; finite element approximation; convergenceReferences:
| [1] | Bartels, S., Numerical analysis of a finite element scheme for the approximation of harmonic maps into surfaces. Math. Comp. 79 (2010), 1263-1301.Zbl1211.65151 MR2629993 · Zbl 1211.65151 |
| [2] | Dierkes, U., Hildebrandt, S., & Sauvigny, F., Minimal Surfaces, second ed., vol. 339 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].Springer, Heidelberg, 2010.Zbl1213.53002 MR2566897 · Zbl 1213.53002 |
| [3] | Dierkes, U., Hildebrandt, S., & Tromba, A. J., Global Analysis of Minimal Surfaces, second ed., vol. 341 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 2010.Zbl1213.53013 MR2778928 · Zbl 1213.53013 |
| [4] | Dierkes, U., Hildebrandt, S., & Tromba, A. J., Regularity of Minimal Surfaces, second ed., vol. 340 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 2010.Zbl1213.53003 MR2760441 · Zbl 1213.53003 |
| [5] | Dziuk, G. & Hutchinson, J. E., The discrete Plateau problem: Algorithm and numerics. Math. Comp. 68 (1999), 1-23.Zbl0913.65062 MR1613695 · Zbl 0913.65062 |
| [6] | Dziuk, G. & Hutchinson, J. E., The discrete Plateau problem: Convergence results. Math. Comp. 68 (1999), 519-546.Zbl1043.65126 MR1613699 · Zbl 1043.65126 |
| [7] | Forster, O., Analysis 2. Vieweg, Braunschweig, 2006.Zbl1106.26001 MR0492102 · Zbl 1106.26001 |
| [8] | Gilbarg, D. & Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, second ed., vol. 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1983.Zbl0562.35001 MR0737190 · Zbl 0562.35001 |
| [9] | Jenschke, T., Analysis und Numerik von Minimalfl¨achen mit freien R¨andern. Dissertation, Universit¨at Duisburg-Essen 2017. |
| [10] | Jenschke, T., Approximation of minimal surfaces with free boundaries: Convergence results. IMA Journal of Numerical Analysis00 (2018), 1-30.Doi 10.1093/imanum/dry043 576U.DIERKES,T.JENSCHKE AND P.POZZI · Zbl 1476.65303 |
| [11] | Lions, J.-L. & Magenes, E.,Non-homogeneous Boundary Value Problems and Applications. Vol. I. Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.Zbl0223.35039 MR0350177 · Zbl 0223.35039 |
| [12] | Morrey, Jr., C. B., Multiple Integrals in the Calculus of Variations. Classics in Mathematics. SpringerVerlag, Berlin, 2008. Reprint of the 1966 edition [MR0202511].Zbl1213.49002 MR2492985 · Zbl 1213.49002 |
| [13] | Nitsche, J. C. C.,Vorlesungen ¨uber Minimalfl¨achen.Springer-Verlag, Berlin-New York, 1975.Die Grundlehren der mathematischen Wissenschaften, Band 199.Zbl0319.53003 MR0448224 |
| [14] | Pozzi, P., L2-estimate for the discrete Plateau problem. Math. Comp. 73 (2004), 1763-1777.Zbl1054. 65066 MR2059735 · Zbl 1054.65066 |
| [15] | Steinhilber, J., Numerical analysis for harmonic maps between hypersurfaces and grid improvement for computational parametric geometric flows. Dissertation, Albert-Ludwigs-Universit¨at Freiburg i. Br. 2014. Zbl1296.53010 · Zbl 1296.53010 |
| [16] | Tchakoutio, P., The Numerical Approximation of Minimal Surfaces with Free Boundaries by Finite Elements. Dissertation, Albert-Ludwigs-Universit¨at Freiburg i. Br. 2002.Zbl1019.65041 · Zbl 1019.65041 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
