Previous |  Up |  Next

Article

Keywords:
perimeter; relative isoperimetric inequality; $p$-Laplacian; eigenfunctions; steepest decent method
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.
References:
[1] H. W. Alt, S. Luckhaus: Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983), 311–341. MR 0706391
[2] A. Cianchi: On relative isoperimetric inequalities in the plane. Bollettino U.M.I. 7 (1989), 3–13. MR 0997998 | Zbl 0674.49030
[3] F. Di Benedetto: Degenerate Parabolic Equations. Springer, Basel, 1993. MR 1230384
[4] P. Drábek, A. Kufner, F. Nicolosi: Quasilinear Elliptic Equations with Degenerations and Singularities. Walter de Gruyter, Berlin, 1997. MR 1460729
[5] H. Federer, W. H. Flemming: Normal and integral currents. Ann. Math. 72 (1960), 458–520. MR 0123260
[6] W. H. Flemming, R. Rishel: An integral formula for total gradient variation. Arch. Math. 11 (1960), 218–222. MR 0114892
[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
[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
[9] H. Gajewski, K. Gröger, K. Zacharias: Nichtlineare Operatorgleichungen ond Operatordifferentialgleichungen. Akademie, Berlin, 1974. MR 0636412
[10] D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order. Springer, 1983. MR 0737190
[11] E. Giusti: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Basel, 1984. MR 0775682 | Zbl 0545.49018
[12] O. Schenk, W. Fichtner, K. Gärtner: ETH-Zürich. Technical Report No. 97/17.
[13] E. Zeidler: Nonlinear functional Analysis and Its Applications II/B. Springer, 1983.
Partner of
EuDML logo