# Article

 Title: An application of eigenfunctions of $p$-Laplacians to domain separation (English) Author: Gajewski, Herbert Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 126 Issue: 2 Year: 2001 Pages: 395-401 Summary lang: English . Category: math . Summary: We are interested in algorithms for constructing surfaces $\Gamma$ of possibly small measure that separate a given domain $\Omega$ into two regions of equal measure. Using the integral formula for the total gradient variation, we show that such separators can be constructed approximatively by means of sign changing eigenfunctions of the $p$-Laplacians, $p \rightarrow 1$, under homogeneous Neumann boundary conditions. These eigenfunctions turn out to be limits of steepest descent methods applied to suitable norm quotients. (English) Keyword: perimeter Keyword: relative isoperimetric inequality Keyword: $p$-Laplacian Keyword: eigenfunctions Keyword: steepest decent method MSC: 35J20 MSC: 35P30 MSC: 58E12 idZBL: Zbl 0979.35041 idMR: MR1844277 DOI: 10.21136/MB.2001.134019 . Date available: 2009-09-24T21:51:50Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/134019 . Reference: [1] H. W. Alt, S. Luckhaus: Quasilinear elliptic-parabolic differential equations.Math. Z. 183 (1983), 311–341. MR 0706391, 10.1007/BF01176474 Reference: [2] A. Cianchi: On relative isoperimetric inequalities in the plane.Bollettino U.M.I. 7 (1989), 3–13. Zbl 0674.49030, MR 0997998 Reference: [3] F. Di Benedetto: Degenerate Parabolic Equations.Springer, Basel, 1993. MR 1230384 Reference: [4] P. Drábek, A. Kufner, F. Nicolosi: Quasilinear Elliptic Equations with Degenerations and Singularities.Walter de Gruyter, Berlin, 1997. MR 1460729 Reference: [5] H. Federer, W. H. Flemming: Normal and integral currents.Ann. Math. 72 (1960), 458–520. MR 0123260, 10.2307/1970227 Reference: [6] W. H. Flemming, R. Rishel: An integral formula for total gradient variation.Arch. Math. 11 (1960), 218–222. MR 0114892, 10.1007/BF01236935 Reference: [7] H. Gajewski, K. Gärtner: On the discretization of van Roosbroeck’s equations with magnetic field.Z. Angew. Math. Mech. 76 (1996), 247–264. MR 1390298, 10.1002/zamm.19960760502 Reference: [8] H. Gajewski, K. Gärtner: Domain separation by means of sign changing eigenfunctions of $p$-Laplacians.Preprint No. 526, Weierstraß Institute, Berlin, 1999. MR 1880955 Reference: [9] H. Gajewski, K. Gröger, K. Zacharias: Nichtlineare Operatorgleichungen ond Operatordifferentialgleichungen.Akademie, Berlin, 1974. MR 0636412 Reference: [10] D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order.Springer, 1983. MR 0737190 Reference: [11] E. Giusti: Minimal Surfaces and Functions of Bounded Variation.Birkhäuser, Basel, 1984. Zbl 0545.49018, MR 0775682 Reference: [12] O. Schenk, W. Fichtner, K. Gärtner: ETH-Zürich.Technical Report No. 97/17. Reference: [13] E. Zeidler: Nonlinear functional Analysis and Its Applications II/B.Springer, 1983. .

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