An application of eigenfunctions of p -Laplacians to domain separation

Herbert Gajewski

Mathematica Bohemica (2001)

  • Volume: 126, Issue: 2, page 395-401
  • ISSN: 0862-7959

Abstract

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We are interested in algorithms for constructing surfaces Γ of possibly small measure that separate a given domain Ω 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 → 1 , under homogeneous Neumann boundary conditions. These eigenfunctions turn out to be limits of steepest descent methods applied to suitable norm quotients.

How to cite

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Gajewski, Herbert. "An application of eigenfunctions of $p$-Laplacians to domain separation." Mathematica Bohemica 126.2 (2001): 395-401. <http://eudml.org/doc/248826>.

@article{Gajewski2001,
abstract = {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.},
author = {Gajewski, Herbert},
journal = {Mathematica Bohemica},
keywords = {perimeter; relative isoperimetric inequality; $p$-Laplacian; eigenfunctions; steepest decent method; perimeter; relative isoperimetric inequality; -Laplacian; eigenfunctions; steepest decent method},
language = {eng},
number = {2},
pages = {395-401},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An application of eigenfunctions of $p$-Laplacians to domain separation},
url = {http://eudml.org/doc/248826},
volume = {126},
year = {2001},
}

TY - JOUR
AU - Gajewski, Herbert
TI - An application of eigenfunctions of $p$-Laplacians to domain separation
JO - Mathematica Bohemica
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 126
IS - 2
SP - 395
EP - 401
AB - 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.
LA - eng
KW - perimeter; relative isoperimetric inequality; $p$-Laplacian; eigenfunctions; steepest decent method; perimeter; relative isoperimetric inequality; -Laplacian; eigenfunctions; steepest decent method
UR - http://eudml.org/doc/248826
ER -

References

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  7. 10.1002/zamm.19960760502, Z. Angew. Math. Mech. 76 (1996), 247–264. (1996) MR1390298DOI10.1002/zamm.19960760502
  8. Domain separation by means of sign changing eigenfunctions of p -Laplacians, Preprint No. 526, Weierstraß Institute, Berlin, 1999. (1999) MR1880955
  9. Nichtlineare Operatorgleichungen ond Operatordifferentialgleichungen, Akademie, Berlin, 1974. (1974) MR0636412
  10. Elliptic Partial Differential Equations of Second Order, Springer, 1983. (1983) MR0737190
  11. Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Basel, 1984. (1984) Zbl0545.49018MR0775682
  12. ETH-Zürich, Technical Report No. 97/17. 
  13. Nonlinear functional Analysis and Its Applications II/B, Springer, 1983. (1983) 

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