Symmetrization of functions and principal eigenvalues of elliptic operators

François Hamel[1]; Nikolai Nadirashvili[2]; Emmanuel Russ[3]

  • [1] Aix-Marseille Université & Institut Universitaire de France LATP Avenue Escadrille Normandie-Niemen 13397 Marseille Cedex 20 France
  • [2] CNRS, LATP 39 rue F. Joliot-Curie 13453 Marseille Cedex 13 France
  • [3] Université Joseph Fourier Institut Fourier 100 rue des Maths BP 74 38402 St-Martin d’Hères France

Séminaire Laurent Schwartz — EDP et applications (2011-2012)

  • Volume: 2011-2012, page 1-15
  • ISSN: 2266-0607

Abstract

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In this paper, we consider shape optimization problems for the principal eigenvalues of second order uniformly elliptic operators in bounded domains of ℝ n . We first recall the classical Rayleigh-Faber-Krahn problem, that is the minimization of the principal eigenvalue of the Dirichlet Laplacian in a domain with fixed Lebesgue measure. We then consider the case of the Laplacian with a bounded drift, that is the operator - Δ + v · ∇ , for which the minimization problem is still well posed. Next, we deal with more general elliptic operators - div ( A ∇ ) + v · ∇ + V , for which the coefficients fulfill various pointwise, integral or geometric constraints. In all cases, some operators with radially symmetric coefficients in an equimeasurable ball are shown to have smaller principal eigenvalues. Whereas the Faber-Krahn proof relies on the classical Schwarz symmetrization, another type of symmetrization is defined to handle the case of general (possibly non-symmetric) operators.

How to cite

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Hamel, François, Nadirashvili, Nikolai, and Russ, Emmanuel. "Symmetrization of functions and principal eigenvalues of elliptic operators." Séminaire Laurent Schwartz — EDP et applications 2011-2012 (2011-2012): 1-15. <http://eudml.org/doc/251151>.

@article{Hamel2011-2012,
abstract = {In this paper, we consider shape optimization problems for the principal eigenvalues of second order uniformly elliptic operators in bounded domains of $\mathbb\{R\}^n$. We first recall the classical Rayleigh-Faber-Krahn problem, that is the minimization of the principal eigenvalue of the Dirichlet Laplacian in a domain with fixed Lebesgue measure. We then consider the case of the Laplacian with a bounded drift, that is the operator $-\Delta +v\cdot \nabla $, for which the minimization problem is still well posed. Next, we deal with more general elliptic operators $-\mathrm\{div\}(A\nabla )+v\cdot \nabla +V$ , for which the coefficients fulfill various pointwise, integral or geometric constraints. In all cases, some operators with radially symmetric coefficients in an equimeasurable ball are shown to have smaller principal eigenvalues. Whereas the Faber-Krahn proof relies on the classical Schwarz symmetrization, another type of symmetrization is defined to handle the case of general (possibly non-symmetric) operators.},
affiliation = {Aix-Marseille Université & Institut Universitaire de France LATP Avenue Escadrille Normandie-Niemen 13397 Marseille Cedex 20 France; CNRS, LATP 39 rue F. Joliot-Curie 13453 Marseille Cedex 13 France; Université Joseph Fourier Institut Fourier 100 rue des Maths BP 74 38402 St-Martin d’Hères France},
author = {Hamel, François, Nadirashvili, Nikolai, Russ, Emmanuel},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {eigenvalue problem; elliptic operator; Rayleigh-Faber-Krahn problem},
language = {eng},
pages = {1-15},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Symmetrization of functions and principal eigenvalues of elliptic operators},
url = {http://eudml.org/doc/251151},
volume = {2011-2012},
year = {2011-2012},
}

TY - JOUR
AU - Hamel, François
AU - Nadirashvili, Nikolai
AU - Russ, Emmanuel
TI - Symmetrization of functions and principal eigenvalues of elliptic operators
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2011-2012
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2011-2012
SP - 1
EP - 15
AB - In this paper, we consider shape optimization problems for the principal eigenvalues of second order uniformly elliptic operators in bounded domains of $\mathbb{R}^n$. We first recall the classical Rayleigh-Faber-Krahn problem, that is the minimization of the principal eigenvalue of the Dirichlet Laplacian in a domain with fixed Lebesgue measure. We then consider the case of the Laplacian with a bounded drift, that is the operator $-\Delta +v\cdot \nabla $, for which the minimization problem is still well posed. Next, we deal with more general elliptic operators $-\mathrm{div}(A\nabla )+v\cdot \nabla +V$ , for which the coefficients fulfill various pointwise, integral or geometric constraints. In all cases, some operators with radially symmetric coefficients in an equimeasurable ball are shown to have smaller principal eigenvalues. Whereas the Faber-Krahn proof relies on the classical Schwarz symmetrization, another type of symmetrization is defined to handle the case of general (possibly non-symmetric) operators.
LA - eng
KW - eigenvalue problem; elliptic operator; Rayleigh-Faber-Krahn problem
UR - http://eudml.org/doc/251151
ER -

References

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