Rearrangement inequalities and applications to isoperimetric problems for eigenvalues

Abstract

Let $\Omega$ be a bounded $C^{2}$ domain in $\Bbb{R}^n$, where $n$ is any positive integer, and let $\Omega^{\ast}$ be the Euclidean ball centered at $0$ and having the same Lebesgue measure as $\Omega$. Consider the operator $L=-\mathrm{div}(A\nabla)+v\cdot \nabla +V$ on $\Omega$ with Dirichlet boundary condition, where the symmetric matrix field $A$ is in $W^{1,\infty}(\Omega)$, the vector field $v$ is in $L^{\infty}(\Omega,\Bbb{R}^n)$ and $V$ is a continuous function in $\overline{\Omega}$. We prove that minimizing the principal eigenvalue of $L$ when the Lebesgue measure of $\Omega$ is fixed and when $A$, $v$ and $V$ vary under some constraints is the same as minimizing the principal eigenvalue of some operators $L^*$ in the ball $\Omega^*$ with smooth and radially symmetric coefficients. The constraints which are satisfied by the original coefficients in $\Omega$ and the new ones in $\Omega^*$ are expressed in terms of some distribution functions or some integral, pointwise or geometric quantities. Some strict comparisons are also established when $\Omega$ is not a ball. To these purposes, we associate to the principal eigenfunction $\varphi$ of $L$ a new symmetric rearrangement defined on $\Omega^*$, which is different from the classical Schwarz symmetrization and which preserves the integral of $\mathrm{div}(A\nabla\varphi)$ on suitable equi-measurable sets. A substantial part of the paper is devoted to the proofs of pointwise and integral inequalities of independent interest which are satisfied by this rearrangement. The comparisons for the eigenvalues hold for general operators of the type $L$ and they are new even for symmetric operators. Furthermore they generalize, in particular, and provide an alternative proof of the well-known Rayleigh-Faber-Krahn isoperimetric inequality about the principal eigenvalue of the Laplacian under Dirichlet boundary condition on a domain with fixed Lebesgue measure.

  • [a] S. Agmon, "On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds," in Methods of Functional Analysis and Theory of Elliptic Equations, Naples: Liguori, 1983, pp. 19-52.
    @incollection {a, MRKEY = {0819005},
      AUTHOR = {Agmon, Shmuel},
      TITLE = {On positivity and decay of solutions of second order elliptic equations on {R}iemannian manifolds},
      BOOKTITLE = {Methods of Functional Analysis and Theory of Elliptic Equations},
      VENUE={{N}aples, 1982},
      PAGES = {19--52},
      PUBLISHER = {Liguori},
      ADDRESS = {Naples},
      YEAR = {1983},
      MRCLASS = {58G25 (35J10)},
      MRNUMBER = {0819005},
      MRREVIEWER = {G. Albinus},
      ZBLNUMBER = {0595.58044},
      }
  • [at1] A. Alvino and G. Trombetti, "Elliptic equations with lower-order terms and reordering," Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., vol. 66, iss. 3, pp. 194-200, 1979.
    @article {at1, MRKEY = {0606083},
      AUTHOR = {Alvino, Angelo and Trombetti, Guido},
      TITLE = {Elliptic equations with lower-order terms and reordering},
      JOURNAL = {Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.},
      FJOURNAL = {Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali. Serie VIII},
      VOLUME = {66},
      YEAR = {1979},
      NUMBER = {3},
      PAGES = {194--200},
      ISSN = {0392-7881},
      CODEN = {AANLAW},
      MRCLASS = {35J15},
      MRNUMBER = {0606083},
      MRREVIEWER = {M. Merigot},
      ZBLNUMBER = {0455.35052},
     }
  • [at2] A. Alvino and G. Trombetti, "On the best majorization constants for a class of degenerate and nondegenerate elliptic equations," Ricerche Mat., vol. 30, iss. 1, pp. 15-33, 1981.
    @article {at2, MRKEY = {0657553},
      AUTHOR = {Alvino, Angelo and Trombetti, Guido},
      TITLE = {On the best majorization constants for a class of degenerate and nondegenerate elliptic equations},
      JOURNAL = {Ricerche Mat.},
      FJOURNAL = {Ricerche di Matematica},
      VOLUME = {30},
      YEAR = {1981},
      NUMBER = {1},
      PAGES = {15--33},
      ISSN = {0035-5038},
      CODEN = {RCMTAE},
      MRCLASS = {35J25 (35J70)},
      MRNUMBER = {0657553},
      MRREVIEWER = {Ya. A. Ro{\u\i}tberg},
      ZBLNUMBER = {0489.35013},
     }
  • [atlm] Go to document A. Alvino, G. Trombetti, P. Lions, and S. Matarasso, "Comparison results for solutions of elliptic problems via symmetrization," Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 16, iss. 2, pp. 167-188, 1999.
    @article {atlm, MRKEY = {1674768},
      AUTHOR = {Alvino, Angelo and Trombetti, Guido and Lions, Pierre-Louis and Matarasso, Silvano},
      TITLE = {Comparison results for solutions of elliptic problems via symmetrization},
      JOURNAL = {Ann. Inst. H. Poincaré Anal. Non Linéaire},
      FJOURNAL = {Annales de l'Institut Henri Poincaré. Analyse Non Linéaire},
      VOLUME = {16},
      YEAR = {1999},
      NUMBER = {2},
      PAGES = {167--188},
      ISSN = {0294-1449},
      MRCLASS = {35J25 (34B15 35B05 49J40)},
      MRNUMBER = {1674768},
      MRREVIEWER = {Lisa Santos},
      DOI = {10.1016/S0294-1449(99)80011-0},
      ZBLNUMBER = {0924.35038},
      }
  • [ashben] Go to document M. S. Ashbaugh and R. D. Benguria, "A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions," Ann. of Math., vol. 135, iss. 3, pp. 601-628, 1992.
    @article {ashben, MRKEY = {1166646},
      AUTHOR = {Ashbaugh, Mark S. and Benguria, Rafael D.},
      TITLE = {A sharp bound for the ratio of the first two eigenvalues of {D}irichlet {L}aplacians and extensions},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {135},
      YEAR = {1992},
      NUMBER = {3},
      PAGES = {601--628},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {35P15 (35J05)},
      MRNUMBER = {1166646},
      MRREVIEWER = {Cornelius O. Horgan},
      DOI = {10.2307/2946578},
      ZBLNUMBER = {0757.35052},
      }
  • [ab2] Go to document M. S. Ashbaugh and R. D. Benguria, "Eigenvalue ratios for Sturm-Liouville operators," J. Differential Equations, vol. 103, iss. 1, pp. 205-219, 1993.
    @article {ab2, MRKEY = {1218744},
      AUTHOR = {Ashbaugh, Mark S. and Benguria, Rafael D.},
      TITLE = {Eigenvalue ratios for {S}turm-{L}iouville operators},
      JOURNAL = {J. Differential Equations},
      FJOURNAL = {Journal of Differential Equations},
      VOLUME = {103},
      YEAR = {1993},
      NUMBER = {1},
      PAGES = {205--219},
      ISSN = {0022-0396},
      CODEN = {JDEQAK},
      MRCLASS = {34L15 (34B24 34L40 47E05)},
      MRNUMBER = {1218744},
      MRREVIEWER = {Horst Behncke},
      DOI = {10.1006/jdeq.1993.1047},
      ZBLNUMBER = {0785.34027},
      }
  • [ab3] M. S. Ashbaugh and R. D. Benguria, "Isoperimetric bounds for higher eigenvalue ratios for the $n$-dimensional fixed membrane problem," Proc. Roy. Soc. Edinburgh Sect. A, vol. 123, iss. 6, pp. 977-985, 1993.
    @article {ab3, MRKEY = {1263898},
      AUTHOR = {Ashbaugh, Mark S. and Benguria, Rafael D.},
      TITLE = {Isoperimetric bounds for higher eigenvalue ratios for the {$n$}-dimensional fixed membrane problem},
      JOURNAL = {Proc. Roy. Soc. Edinburgh Sect. A},
      FJOURNAL = {Proceedings of the Royal Society of Edinburgh. Section A. Mathematics},
      VOLUME = {123},
      YEAR = {1993},
      NUMBER = {6},
      PAGES = {977--985},
      ISSN = {0308-2105},
      CODEN = {PEAMDU},
      MRCLASS = {35P15},
      MRNUMBER = {1263898},
      MRREVIEWER = {G{ü}nter Berger},
      ZBLNUMBER = {0819.35104},
      }
  • [ashbenclamped] Go to document M. S. Ashbaugh and R. D. Benguria, "On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions," Duke Math. J., vol. 78, iss. 1, pp. 1-17, 1995.
    @article {ashbenclamped, MRKEY = {1328749},
      AUTHOR = {Ashbaugh, Mark S. and Benguria, Rafael D.},
      TITLE = {On {R}ayleigh's conjecture for the clamped plate and its generalization to three dimensions},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {78},
      YEAR = {1995},
      NUMBER = {1},
      PAGES = {1--17},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {35P15 (35J40 47F05 73K10)},
      MRNUMBER = {1328749},
      MRREVIEWER = {R. Kodn{á}r},
      DOI = {10.1215/S0012-7094-95-07801-6},
      ZBLNUMBER = {0833.35035},
      }
  • [ahs] Go to document M. S. Ashbaugh, E. M. Harrell II, and R. Svirsky, "On minimal and maximal eigenvalue gaps and their causes," Pacific J. Math., vol. 147, iss. 1, pp. 1-24, 1991.
    @article {ahs, MRKEY = {1081670},
      AUTHOR = {Ashbaugh, Mark S. and Harrell, II, Evans M. and Svirsky, Roman},
      TITLE = {On minimal and maximal eigenvalue gaps and their causes},
      JOURNAL = {Pacific J. Math.},
      FJOURNAL = {Pacific Journal of Mathematics},
      VOLUME = {147},
      YEAR = {1991},
      NUMBER = {1},
      PAGES = {1--24},
      ISSN = {0030-8730},
      CODEN = {PJMAAI},
      MRCLASS = {35P15 (35J10 81Q10)},
      MRNUMBER = {1081670},
      MRREVIEWER = {E. M{ü}ller-Pfeiffer},
      URL = {http://projecteuclid.org/getRecord?id=euclid.pjm/1102645038},
      ZBLNUMBER = {0734.35061},
      }
  • [bandle1] Go to document C. Bandle, "Isoperimetric inequality for some eigenvalues of an inhomogeneous, free membrane," SIAM J. Appl. Math., vol. 22, pp. 142-147, 1972.
    @article {bandle1, MRKEY = {0313648},
      AUTHOR = {Bandle, Catherine},
      TITLE = {Isoperimetric inequality for some eigenvalues of an inhomogeneous, free membrane},
      JOURNAL = {SIAM J. Appl. Math.},
      FJOURNAL = {SIAM Journal on Applied Mathematics},
      VOLUME = {22},
      YEAR = {1972},
      PAGES = {142--147},
      ISSN = {0036-1399},
      MRCLASS = {35P15 (52A40 73.35)},
      MRNUMBER = {0313648},
      MRREVIEWER = {D. O. Banks},
      DOI = {10.1137/0122016},
      ZBLNUMBER = {0237.35069},
      }
  • [bandle2] C. Bandle, Isoperimetric Inequalities and Applications, Boston, MA: Pitman (Advanced Publishing Program), 1980, vol. 7.
    @book {bandle2, MRKEY = {0572958},
      AUTHOR = {Bandle, Catherine},
      TITLE = {Isoperimetric Inequalities and Applications},
      SERIES = {Monogr. Studies in Math.},
      VOLUME = {7},
      PUBLISHER = {Pitman (Advanced Publishing Program)},
      ADDRESS = {Boston, MA},
      YEAR = {1980},
      PAGES = {x+228},
      ISBN = {0-273-08423-2},
      MRCLASS = {35P15 (49G05 52A40 58G20)},
      MRNUMBER = {0572958},
      MRREVIEWER = {H. Hochstadt},
      ZBLNUMBER = {0436.35063},
      }
  • [barnes] Go to document D. C. Barnes, "Extremal problems for eigenvalues with applications to buckling, vibration and sloshing," SIAM J. Math. Anal., vol. 16, iss. 2, pp. 341-357, 1985.
    @article {barnes, MRKEY = {0777472},
      AUTHOR = {Barnes, David C.},
      TITLE = {Extremal problems for eigenvalues with applications to buckling, vibration and sloshing},
      JOURNAL = {SIAM J. Math. Anal.},
      FJOURNAL = {SIAM Journal on Mathematical Analysis},
      VOLUME = {16},
      YEAR = {1985},
      NUMBER = {2},
      PAGES = {341--357},
      ISSN = {0036-1410},
      CODEN = {SJMAAH},
      MRCLASS = {49G05 (34B25)},
      MRNUMBER = {0777472},
      MRREVIEWER = {Jan Bochenek},
      DOI = {10.1137/0516025},
      ZBLNUMBER = {0568.73047},
      }
  • [bhn] Go to document H. Berestycki, F. Hamel, and N. Nadirashvili, "Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena," Comm. Math. Phys., vol. 253, iss. 2, pp. 451-480, 2005.
    @article {bhn, MRKEY = {2140256},
      AUTHOR = {Berestycki, Henri and Hamel, Fran{ç}ois and Nadirashvili, Nikolai},
      TITLE = {Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {253},
      YEAR = {2005},
      NUMBER = {2},
      PAGES = {451--480},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {35J15 (35B45 35K57 35P05 35P15)},
      MRNUMBER = {2140256},
      MRREVIEWER = {Jes{ú}s Hern{á}ndez},
      DOI = {10.1007/s00220-004-1201-9},
      ZBLNUMBER = {1123.35033},
      }
  • [bnv] Go to document H. Berestycki, L. Nirenberg, and S. R. S. Varadhan, "The principal eigenvalue and maximum principle for second-order elliptic operators in general domains," Comm. Pure Appl. Math., vol. 47, iss. 1, pp. 47-92, 1994.
    @article {bnv, MRKEY = {1258192},
      AUTHOR = {Berestycki, H. and Nirenberg, L. and Varadhan, S. R. S.},
      TITLE = {The principal eigenvalue and maximum principle for second-order elliptic operators in general domains},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {47},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {47--92},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {35J15 (35B50 35J20 35P15)},
      MRNUMBER = {1258192},
      MRREVIEWER = {M. S. Agranovich},
      DOI = {10.1002/cpa.3160470105},
      ZBLNUMBER = {0806.35129},
      }
  • [bossel] M. Bossel, "Membranes élastiquement liées: extension du théorème de Rayleigh-Faber-Krahn et de l’inégalité de Cheeger," C. R. Acad. Sci. Paris Sér. I Math., vol. 302, iss. 1, pp. 47-50, 1986.
    @article {bossel, MRKEY = {0827106},
      AUTHOR = {Bossel, Marie-H{é}l{è}ne},
      TITLE = {Membranes élastiquement liées: extension du théorème de {R}ayleigh-{F}aber-{K}rahn et de l'inégalité de {C}heeger},
      JOURNAL = {C. R. Acad. Sci. Paris Sér. I Math.},
      FJOURNAL = {Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique},
      VOLUME = {302},
      YEAR = {1986},
      NUMBER = {1},
      PAGES = {47--50},
      ISSN = {0249-6291},
      CODEN = {CASMEI},
      MRCLASS = {35P15 (35J05 49H05 58G25)},
      MRNUMBER = {0827106},
      MRREVIEWER = {Catherine Bandle},
      ZBLNUMBER = {0606.73018},
     }
  • [brock] F. Brock, "An isoperimetric inequality for eigenvalues of the Stekloff problem," ZAMM Z. Angew. Math. Mech., vol. 81, iss. 1, pp. 69-71, 2001.
    @article {brock, MRKEY = {1808500},
      AUTHOR = {Brock, F.},
      TITLE = {An isoperimetric inequality for eigenvalues of the {S}tekloff problem},
      JOURNAL = {ZAMM Z. Angew. Math. Mech.},
      FJOURNAL = {ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik. Journal of Applied Mathematics and Mechanics},
      VOLUME = {81},
      YEAR = {2001},
      NUMBER = {1},
      PAGES = {69--71},
      ISSN = {0044-2267},
      MRCLASS = {35P15 (35J05 35J20)},
      MRNUMBER = {1808500},
      ZBLNUMBER = {0971.35055},
      }
  • [bucurhenrot] Go to document D. Bucur and A. Henrot, "Minimization of the third eigenvalue of the Dirichlet Laplacian," R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., vol. 456, iss. 1996, pp. 985-996, 2000.
    @article {bucurhenrot, MRKEY = {1805088},
      AUTHOR = {Bucur, Dorin and Henrot, Antoine},
      TITLE = {Minimization of the third eigenvalue of the {D}irichlet {L}aplacian},
      JOURNAL = {R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.},
      FJOURNAL = {The Royal Society of London. Proceedings. Series A. Mathematical, Physical and Engineering Sciences},
      VOLUME = {456},
      YEAR = {2000},
      NUMBER = {1996},
      PAGES = {985--996},
      ISSN = {1364-5021},
      MRCLASS = {35P15 (35J05 47F05 49R50)},
      MRNUMBER = {1805088},
      MRREVIEWER = {G{ü}nter Berger},
      DOI = {10.1098/rspa.2000.0546},
      ZBLNUMBER = {0974.35082},
      }
  • [co] Go to document S. Cheng and K. Oden, "Isoperimetric inequalities and the gap between the first and second eigenvalues of an Euclidean domain," J. Geom. Anal., vol. 7, iss. 2, pp. 217-239, 1997.
    @article {co, MRKEY = {1646764},
      AUTHOR = {Cheng, Shiu-Yuen and Oden, Kevin},
      TITLE = {Isoperimetric inequalities and the gap between the first and second eigenvalues of an {E}uclidean domain},
      JOURNAL = {J. Geom. Anal.},
      FJOURNAL = {The Journal of Geometric Analysis},
      VOLUME = {7},
      YEAR = {1997},
      NUMBER = {2},
      PAGES = {217--239},
      ISSN = {1050-6926},
      MRCLASS = {35P15 (35J05 58G25)},
      MRNUMBER = {1646764},
      MRREVIEWER = {Norbert Weck},
      ZBLNUMBER = {0920.58054},
      DOI={10.1007/BF02921721},
      }
  • [d] Go to document D. Daners, "A Faber-Krahn inequality for Robin problems in any space dimension," Math. Ann., vol. 335, iss. 4, pp. 767-785, 2006.
    @article {d, MRKEY = {2232016},
      AUTHOR = {Daners, Daniel},
      TITLE = {A {F}aber-{K}rahn inequality for {R}obin problems in any space dimension},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {335},
      YEAR = {2006},
      NUMBER = {4},
      PAGES = {767--785},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {35P15 (35J25)},
      MRNUMBER = {2232016},
      MRREVIEWER = {Siegfried Carl},
      DOI = {10.1007/s00208-006-0753-8},
      ZBLNUMBER = {05046891},
      }
  • [dk] Go to document D. Daners and J. Kennedy, "Uniqueness in the Faber-Krahn inequality for Robin problems," SIAM J. Math. Anal., vol. 39, iss. 4, pp. 1191-1207, 2007/08.
    @article {dk, MRKEY = {2368899},
      AUTHOR = {Daners, Daniel and Kennedy, James},
      TITLE = {Uniqueness in the {F}aber-{K}rahn inequality for {R}obin problems},
      JOURNAL = {SIAM J. Math. Anal.},
      FJOURNAL = {SIAM Journal on Mathematical Analysis},
      VOLUME = {39},
      YEAR = {2007/08},
      NUMBER = {4},
      PAGES = {1191--1207},
      ISSN = {0036-1410},
      MRCLASS = {35J05 (35J25 35P15)},
      MRNUMBER = {2368899},
      MRREVIEWER = {Srinivasan Kesavan},
      DOI = {10.1137/060675629},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {1155.35403},
      }
  • [Faber] G. Faber, "Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt," Sitzungsberichte der mathematisch-physikalischen Klasse der Bauerischen Akademie der Wissenschaften zu München, pp. 169-172, 1923.
    @article{Faber,
      author={Faber, G.},
      TITLE={Beweis, dass unter allen homogenen {M}embranen von gleicher {F}läche und gleicher {S}pannung die kreisförmige den tiefsten {G}rundton gibt},
      JOURNAL={Sitzungsberichte der mathematisch-physikalischen Klasse der Bauerischen Akademie der Wissenschaften zu München},
      YEAR={1923},
      PAGES={169--172},
      jfmnumber={49.0342.03},
     }
  • [fr] Go to document A. Friedman, "The asymptotic behavior of the first real eigenvalue of a second order elliptic operator with a small parameter in the highest derivatives," Indiana Univ. Math. J., vol. 22, pp. 1005-1015, 1972/73.
    @article {fr, MRKEY = {0320551},
      AUTHOR = {Friedman, Avner},
      TITLE = {The asymptotic behavior of the first real eigenvalue of a second order elliptic operator with a small parameter in the highest derivatives},
      JOURNAL = {Indiana Univ. Math. J.},
      FJOURNAL = {Indiana University Mathematics Journal},
      VOLUME = {22},
      YEAR = {1972/73},
      PAGES = {1005--1015},
      ISSN = {0022-2518},
      MRCLASS = {35P20},
      MRNUMBER = {0320551},
      MRREVIEWER = {P. Szeptycki},
      DOI = {10.1512/iumj.1973.22.22084},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0245.35065},
      }
  • [hnr] Go to document F. Hamel, N. Nadirashvili, and E. Russ, "An isoperimetric inequality for the principal eigenvalue of the Laplacian with drift," C. R. Math. Acad. Sci. Paris, vol. 340, iss. 5, pp. 347-352, 2005.
    @article {hnr, MRKEY = {2127108},
      AUTHOR = {Hamel, Fran{ç}ois and Nadirashvili, Nikolai and Russ, Emmanuel},
      TITLE = {An isoperimetric inequality for the principal eigenvalue of the {L}aplacian with drift},
      JOURNAL = {C. R. Math. Acad. Sci. Paris},
      FJOURNAL = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
      VOLUME = {340},
      YEAR = {2005},
      NUMBER = {5},
      PAGES = {347--352},
      ISSN = {1631-073X},
      MRCLASS = {35J25 (35P15)},
      MRNUMBER = {2127108},
      DOI = {10.1016/j.crma.2005.01.012},
      ZBLNUMBER = {1066.35066},
      }
  • [hnrpreprint] Go to document F. Hamel, N. Nadirashvili, and E. Russ, A Faber-Krahn inequality with drift.
    @misc{hnrpreprint,
      author={Hamel, Fran{ç}ois and Nadirashvili, Nikolai and Russ, Emmanuel},
      TITLE = {A {F}aber-{K}rahn inequality with drift},
      NOTE={preprint},
      URL={http://fr.arxiv.org/abs/math.AP/0607585},
      }
  • [henrot] Go to document A. Henrot, "Minimization problems for eigenvalues of the Laplacian," J. Evol. Equ., vol. 3, iss. 3, pp. 443-461, 2003.
    @article {henrot, MRKEY = {2019029},
      AUTHOR = {Henrot, Antoine},
      TITLE = {Minimization problems for eigenvalues of the {L}aplacian},
      JOURNAL = {J. Evol. Equ.},
      FJOURNAL = {Journal of Evolution Equations},
      VOLUME = {3},
      YEAR = {2003},
      NUMBER = {3},
      PAGES = {443--461},
      ISSN = {1424-3199},
      MRCLASS = {49Q10 (35P15 49J20 49R50)},
      MRNUMBER = {2019029},
      MRREVIEWER = {E. Lami Dozo},
      DOI = {10.1007/s00028-003-0111-0},
      ZBLNUMBER = {1049.49029},
      }
  • [henrot2] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Basel: Birkhäuser, 2006.
    @book {henrot2, MRKEY = {2251558},
      AUTHOR = {Henrot, Antoine},
      TITLE = {Extremum Problems for Eigenvalues of Elliptic Operators},
      SERIES = {Frontiers in Mathematics},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {2006},
      PAGES = {x+202},
      ISBN = {978-3-7643-7705-2; 3-7643-7705-4},
      MRCLASS = {35P05 (35J05 35J15 35P15 47F05 49R50)},
      MRNUMBER = {2251558},
      MRREVIEWER = {G{ü}nter Berger},
      ZBLNUMBER = {1109.35081},
      }
  • [holland] Go to document C. J. Holland, "A minimum principle for the principal eigenvalue for second-order linear elliptic equations with natural boundary conditions," Comm. Pure Appl. Math., vol. 31, iss. 4, pp. 509-519, 1978.
    @article {holland, MRKEY = {0466940},
      AUTHOR = {Holland, Charles J.},
      TITLE = {A minimum principle for the principal eigenvalue for second-order linear elliptic equations with natural boundary conditions},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {31},
      YEAR = {1978},
      NUMBER = {4},
      PAGES = {509--519},
      ISSN = {0010-3640},
      MRCLASS = {35J20},
      MRNUMBER = {0466940},
      MRREVIEWER = {Z. C. Motteler},
      DOI = {10.1002/cpa.3160310406},
      ZBLNUMBER = {0388.35053},
      }
  • [karaa] Go to document S. Karaa, "Extremal eigenvalue gaps for the Schrödinger operator with Dirichlet boundary conditions," J. Math. Phys., vol. 39, iss. 4, pp. 2325-2332, 1998.
    @article {karaa, MRKEY = {1614731},
      AUTHOR = {Karaa, Samir},
      TITLE = {Extremal eigenvalue gaps for the {S}chrödinger operator with {D}irichlet boundary conditions},
      JOURNAL = {J. Math. Phys.},
      FJOURNAL = {Journal of Mathematical Physics},
      VOLUME = {39},
      YEAR = {1998},
      NUMBER = {4},
      PAGES = {2325--2332},
      ISSN = {0022-2488},
      CODEN = {JMAPAQ},
      MRCLASS = {35P15 (35J10)},
      MRNUMBER = {1614731},
      DOI={10.1063/1.532290},
      ZBLNUMBER={1042.35601},
      }
  • [karaa2] Go to document S. Karaa, "Sharp estimates for the eigenvalues of some differential equations," SIAM J. Math. Anal., vol. 29, iss. 5, pp. 1279-1300, 1998.
    @article {karaa2, MRKEY = {1628259},
      AUTHOR = {Karaa, Samir},
      TITLE = {Sharp estimates for the eigenvalues of some differential equations},
      JOURNAL = {SIAM J. Math. Anal.},
      FJOURNAL = {SIAM Journal on Mathematical Analysis},
      VOLUME = {29},
      YEAR = {1998},
      NUMBER = {5},
      PAGES = {1279--1300},
      ISSN = {0036-1410},
      MRCLASS = {34L15 (34B05 49R05)},
      MRNUMBER = {1628259},
      MRREVIEWER = {Bernard J. Harris},
      DOI = {10.1137/S0036141096307849},
      ZBLNUMBER={0914.34072},
      }
  • [Krahndim2] Go to document E. Krahn, "Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises," Math. Ann., vol. 94, iss. 1, pp. 97-100, 1925.
    @article {Krahndim2, MRKEY = {1512244},
      AUTHOR = {Krahn, E.},
      TITLE = {Über eine von {R}ayleigh formulierte {M}inimaleigenschaft des {K}reises},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {94},
      YEAR = {1925},
      NUMBER = {1},
      PAGES = {97--100},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {Contributed Item},
      MRNUMBER = {1512244},
      DOI = {10.1007/BF01208645},
      jfmnumber = {51.0356.05},
     }
  • [Krahndimn] E. Krahn, "Über Minimaleigenschaft der Kugel in drei und mehr Dimensionen," Acta Comm. Univ. Tartu (Dorpat), vol. A9, pp. 1-44, 1926.
    @article{Krahndimn,
      author={Krahn, E.},
      TITLE={Über Minimaleigenschaft der {K}ugel in drei und mehr Dimensionen},
      JOURNAL={Acta Comm. Univ. Tartu (Dorpat)},
      VOLUME={A9},
      YEAR={1926},
      PAGES={1--44},
      jfmnumber={52.0510.03},
      }
  • [Krahndimnenglish] Edgar Krahn, 1894–1961, A Centenary Volume, Lumiste, &. and Peetre, J., Eds., Amsterdam: IOS Press, 1994.
    @book {Krahndimnenglish, MRKEY = {1298188},
      TITLE = {Edgar {K}rahn, 1894--1961, A {C}entenary {V}olume},
      EDITOR={{Ü}lo Lumiste and Jaak Peetre},
      PUBLISHER = {IOS Press},
      ADDRESS = {Amsterdam},
      YEAR = {1994},
      PAGES = {x+196},
      ISBN = {90-5199-168-1},
      MRCLASS = {01A70 (01-06)},
      MRNUMBER = {1298188},
      ZBLNUMBER={0834.01010},
      }
  • [ly] Go to document P. Li and S. -T. Yau, "On the Schrödinger equation and the eigenvalue problem," Comm. Math. Phys., vol. 88, iss. 3, pp. 309-318, 1983.
    @article {ly, MRKEY = {0701919},
      AUTHOR = {Li, Peter and Yau, S.-T.},
      TITLE = {On the {S}chrödinger equation and the eigenvalue problem},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {88},
      YEAR = {1983},
      NUMBER = {3},
      PAGES = {309--318},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {58G25 (35P99 58G05 81C05)},
      MRNUMBER = {0701919},
      MRREVIEWER = {Harold Donnelly},
      ZBLNUMBER={0554.34029},
      DOI = {10.1007/BF01213210},
     }
  • [m] Go to document P. Marcellini, "Bounds for the third membrane eigenvalue," J. Differential Equations, vol. 37, iss. 3, pp. 438-443, 1980.
    @article {m, MRKEY = {0590001},
      AUTHOR = {Marcellini, Paolo},
      TITLE = {Bounds for the third membrane eigenvalue},
      JOURNAL = {J. Differential Equations},
      FJOURNAL = {Journal of Differential Equations},
      VOLUME = {37},
      YEAR = {1980},
      NUMBER = {3},
      PAGES = {438--443},
      ISSN = {0022-0396},
      CODEN = {JDEQAK},
      MRCLASS = {35P15 (49G05)},
      MRNUMBER = {0590001},
      MRREVIEWER = {J. Hersch},
      DOI = {10.1016/0022-0396(80)90108-4},
      ZBLNUMBER = {0421.35066},
      }
  • [nad] Go to document N. S. Nadirashvili, "Rayleigh’s conjecture on the principal frequency of the clamped plate," Arch. Rational Mech. Anal., vol. 129, iss. 1, pp. 1-10, 1995.
    @article {nad, MRKEY = {1328469},
      AUTHOR = {Nadirashvili, Nikolai S.},
      TITLE = {Rayleigh's conjecture on the principal frequency of the clamped plate},
      JOURNAL = {Arch. Rational Mech. Anal.},
      FJOURNAL = {Archive for Rational Mechanics and Analysis},
      VOLUME = {129},
      YEAR = {1995},
      NUMBER = {1},
      PAGES = {1--10},
      ISSN = {0003-9527},
      CODEN = {AVRMAW},
      MRCLASS = {35P15 (35J40 47F05 73K10)},
      MRNUMBER = {1328469},
      MRREVIEWER = {R. Kodn{á}r},
      DOI = {10.1007/BF00375124},
      ZBLNUMBER = {0826.73035},
      }
  • [np] Go to document R. D. Nussbaum and Y. Pinchover, "On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications," J. Anal. Math., vol. 59, pp. 161-177, 1992.
    @article {np, MRKEY = {1226957},
      AUTHOR = {Nussbaum, Roger D. and Pinchover, Yehuda},
      TITLE = {On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications},
      JOURNAL = {J. Anal. Math.},
      FJOURNAL = {Journal d'Analyse Mathématique},
      VOLUME = {59},
      YEAR = {1992},
      PAGES = {161--177},
      ISSN = {0021-7670},
      CODEN = {JOAMAV},
      MRCLASS = {35J20 (35P05)},
      MRNUMBER = {1226957},
      MRREVIEWER = {Jan Bochenek},
      DOI = {10.1007/BF02790223},
      ZBLNUMBER = {0816.35095},
      }
  • [o] Go to document R. Osserman, "The isoperimetric inequality," Bull. Amer. Math. Soc., vol. 84, iss. 6, pp. 1182-1238, 1978.
    @article {o, MRKEY = {0500557},
      AUTHOR = {Osserman, Robert},
      TITLE = {The isoperimetric inequality},
      JOURNAL = {Bull. Amer. Math. Soc.},
      FJOURNAL = {Bulletin of the American Mathematical Society},
      VOLUME = {84},
      YEAR = {1978},
      NUMBER = {6},
      PAGES = {1182--1238},
      ISSN = {0002-9904},
      MRCLASS = {52A40 (35P15 53A05 53C45)},
      MRNUMBER = {0500557},
      MRREVIEWER = {H. W. Guggenheimer},
      DOI = {10.1090/S0002-9904-1978-14553-4},
      ZBLNUMBER = {0411.52006},
      }
  • [ppw] L. E. Payne, G. Pólya, and H. F. Weinberger, "On the ratio of consecutive eigenvalues," J. Math. and Phys., vol. 35, pp. 289-298, 1956.
    @article {ppw, MRKEY = {0084696},
      AUTHOR = {Payne, L. E. and P{ó}lya, G. and Weinberger, H. F.},
      TITLE = {On the ratio of consecutive eigenvalues},
      JOURNAL = {J. Math. and Phys.},
      FJOURNAL = {Journal of Mathematics and Physics},
      VOLUME = {35},
      YEAR = {1956},
      PAGES = {289--298},
      ISSN = {0097-1421},
      MRCLASS = {35.0X},
      MRNUMBER = {0084696},
      MRREVIEWER = {E. C. Titchmarsh},
      ZBLNUMBER = {0073.08203},
      }
  • [polya] Go to document G. Pólya, "On the characteristic frequencies of a symmetric membrane," Math. Z., vol. 63, pp. 331-337, 1955.
    @article {polya, MRKEY = {0073047},
      AUTHOR = {P{ó}lya, G.},
      TITLE = {On the characteristic frequencies of a symmetric membrane},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {63},
      YEAR = {1955},
      PAGES = {331--337},
      ISSN = {0025-5874},
      MRCLASS = {35.0X},
      MRNUMBER = {0073047},
      MRREVIEWER = {H. F. Weinberger},
      DOI = {10.1007/BF01187944},
      ZBLNUMBER = {0065.08703},
      }
  • [polya2] Go to document G. Pólya, "On the eigenvalues of vibrating membranes," Proc. London Math. Soc., vol. 11, pp. 419-433, 1961.
    @article {polya2, MRKEY = {0129219},
      AUTHOR = {P{ó}lya, G.},
      TITLE = {On the eigenvalues of vibrating membranes},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {11},
      YEAR = {1961},
      PAGES = {419--433},
      ISSN = {0024-6115},
      MRCLASS = {73.35 (35.80)},
      MRNUMBER = {0129219},
      MRREVIEWER = {H. F. Weinberger},
      DOI = {10.1112/plms/s3-11.1.419},
      ZBLNUMBER = {0107.41805},
      }
  • [polyaszego] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton, NJ: Princeton Univ. Press, 1951.
    @book {polyaszego, MRKEY = {0043486},
      AUTHOR = {P{ó}lya, G. and Szeg{ö},
      G.},
      TITLE = {Isoperimetric {I}nequalities in {M}athematical {P}hysics},
      SERIES = {Annals of Mathematics Studies, {\rm No.} 27},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1951},
      PAGES = {xvi+279},
      MRCLASS = {52.0X},
      MRNUMBER = {0043486},
      MRREVIEWER = {M. Brelot},
      ZBLNUMBER = {0044.38301},
      }
  • [Rayleigh] J. W. S. Rayleigh, The Theory of Sound, New York, N. Y.: Dover Publications, 1945.
    @book {Rayleigh, MRKEY = {0016009},
      AUTHOR = {Rayleigh, John William Strutt},
      TITLE = {The {T}heory of {S}ound},
      NOTE = {2nd ed., revised and enlarged (in 2 vols.)},
      PUBLISHER = {Dover Publications},
      ADDRESS={ New York, N. Y.},
      YEAR = {1945},
      PAGES = {Vol. I, xlii+480 pp.; vol. II, xii+504},
      MRCLASS = {76.1X},
      MRNUMBER = {0016009},
      ZBLNUMBER = {0061.45904},
      }
  • [soucek] J. Souvcek and V. Souvcek, "Morse-Sard theorem for real-analytic functions," Comment. Math. Univ. Carolinae, vol. 13, pp. 45-51, 1972.
    @article {soucek, MRKEY = {0308345},
      AUTHOR = {Sou{\v{c}}ek, J. and Sou{\v{c}}ek, V.},
      TITLE = {Morse-{S}ard theorem for real-analytic functions},
      JOURNAL = {Comment. Math. Univ. Carolinae},
      FJOURNAL = {Commentationes Mathematicae Universitatis Carolinae},
      VOLUME = {13},
      YEAR = {1972},
      PAGES = {45--51},
      ISSN = {0010-2628},
      MRCLASS = {26A90},
      MRNUMBER = {0308345},
      MRREVIEWER = {J. W. Neuberger},
      ZBLNUMBER = {0235.26012},
      }
  • [sz] G. Szegö, "Inequalities for certain eigenvalues of a membrane of given area," J. Rational Mech. Anal., vol. 3, pp. 343-356, 1954.
    @article {sz, MRKEY = {0061749},
      AUTHOR = {Szeg{ö},
      G.},
      TITLE = {Inequalities for certain eigenvalues of a membrane of given area},
      JOURNAL = {J. Rational Mech. Anal.},
      VOLUME = {3},
      YEAR = {1954},
      PAGES = {343--356},
      MRCLASS = {36.0X},
      MRNUMBER = {0061749},
      MRREVIEWER = {E. T. Copson},
      ZBLNUMBER = {0055.08802},
      }
  • [talenti] G. Talenti, "Linear elliptic p.d.e.’s: level sets, rearrangements and a priori estimates of solutions," Boll. Un. Mat. Ital. B, vol. 4, iss. 3, pp. 917-949, 1985.
    @article {talenti, MRKEY = {0831299},
      AUTHOR = {Talenti, Giorgio},
      TITLE = {Linear elliptic p.d.e.'s: level sets, rearrangements and a priori estimates of solutions},
      JOURNAL = {Boll. Un. Mat. Ital. B},
      FJOURNAL = {Unione Matematica Italiana. Bollettino. B. Serie VI},
      VOLUME = {4},
      YEAR = {1985},
      NUMBER = {3},
      PAGES = {917--949},
      CODEN = {BLUMAM},
      MRCLASS = {35B05 (35B45 35J35)},
      MRNUMBER = {0831299},
      MRREVIEWER = {Catherine Bandle},
      ZBLNUMBER={0602.35025},
      }
  • [tv] Go to document G. Trombetti and J. L. Vázquez, "A symmetrization result for elliptic equations with lower-order terms," Ann. Fac. Sci. Toulouse Math., vol. 7, iss. 2, pp. 137-150, 1985.
    @article {tv, MRKEY = {0842766},
      AUTHOR = {Trombetti, Guido and V{á}zquez, Juan Luis},
      TITLE = {A symmetrization result for elliptic equations with lower-order terms},
      JOURNAL = {Ann. Fac. Sci. Toulouse Math.},
      FJOURNAL = {Toulouse. Faculté des Sciences. Annales. Mathématiques. Série 5},
      VOLUME = {7},
      YEAR = {1985},
      NUMBER = {2},
      PAGES = {137--150},
      ISSN = {0240-2955},
      MRCLASS = {35J25},
      MRNUMBER = {0842766},
      MRREVIEWER = {Catherine Bandle},
      URL = {http://www.numdam.org/item?id=AFST_1985_5_7_2_137_0},
      ZBLNUMBER = {0617.35036},
      }
  • [v] A. D. Ventcel, "On the asymptotic behavior of the greatest eigenvalue of a second-order elliptic differential operator with a small parameter in the higher derivatives," Soviet Math., vol. 13, pp. 13-17, 1972.
    @article{v,
      author={Ventcel, A. D.},
      TITLE={On the asymptotic behavior of the greatest eigenvalue of a second-order elliptic differential operator with a small parameter in the higher derivatives},
      JOURNAL={Soviet Math.},
      VOLUME={13},
      YEAR={1972},
      PAGES={13--17},
      ZBLNUMBER={0241.35060},
      }
  • [wein] H. F. Weinberger, "An isoperimetric inequality for the $N$-dimensional free membrane problem," J. Rational Mech. Anal., vol. 5, pp. 633-636, 1956.
    @article {wein, MRKEY = {0079286},
      AUTHOR = {Weinberger, H. F.},
      TITLE = {An isoperimetric inequality for the {$N$}-dimensional free membrane problem},
      JOURNAL = {J. Rational Mech. Anal.},
      VOLUME = {5},
      YEAR = {1956},
      PAGES = {633--636},
      MRCLASS = {52.0X},
      MRNUMBER = {0079286},
      MRREVIEWER = {P. Funk},
      ZBLNUMBER = {0071.09902},
      }
  • [wk] Go to document S. A. Wolf and J. B. Keller, "Range of the first two eigenvalues of the Laplacian," Proc. Roy. Soc. London Ser. A, vol. 447, iss. 1930, pp. 397-412, 1994.
    @article {wk, MRKEY = {1312811},
      AUTHOR = {Wolf, Sven Andreas and Keller, Joseph B.},
      TITLE = {Range of the first two eigenvalues of the {L}aplacian},
      JOURNAL = {Proc. Roy. Soc. London Ser. A},
      FJOURNAL = {Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences},
      VOLUME = {447},
      YEAR = {1994},
      NUMBER = {1930},
      PAGES = {397--412},
      ISSN = {0962-8444},
      CODEN = {PRLAAZ},
      MRCLASS = {35P15 (58G25)},
      MRNUMBER = {1312811},
      DOI = {10.1098/rspa.1994.0147},
      ZBLNUMBER = {0816.35097},
      }

Authors

François Hamel

Université Aix-Marseille III
LATP
13997 Marseille
France

and

Institut Universitaire de France

Nikolai Nadirashvili

CNRS et Université d'Aix-Marseille 1 - LATP
Centre de Mathématiques et Informatique
13453 Marseille
France

Emmanuel Russ

Université Aix-Marseille III
LATP
13397 Marseille
France

Current address:

Université Joseph Fourier Grenoble
Institut Fourier
38402 St Martin d'Hères Cedex,
France