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2013 | 1 | 31-35
Tytuł artykułu

Conformal Geometry and the Composite Membrane Problem

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We show that a certain eigenvalue minimization problem in two dimensions for the Laplace operator in conformal classes is equivalent to the composite membrane problem. We again establish such a link in higher dimensions for eigenvalue problems stemming from the critical GJMS operators. New free boundary problems of unstable type arise in higher dimensions linked to the critical GJMS operator. In dimension four, the critical GJMS operator is exactly the Paneitz operator.
Wydawca
Rocznik
Tom
1
Strony
31-35
Opis fizyczny
Daty
otrzymano
2012-09-27
zaakceptowano
2012-12-13
online
2013-01-04
Twórcy
  • Department of Mathematics, Rutgers University,
    110 Frelinghuysen Rd., Piscataway, NJ 08854, U.S.A., chanillo@math.rutgers.edu
Bibliografia
  • Baum, H. and Juhl, A : Conformal Differential Geometry: Q-Curvature and Conformal Holonomy, OberwolfachSeminars 40, 2010, Birkhäuser Verlag.[WoS]
  • Blank, Ivan: Eliminating Mixed Asymptotics in Obstacle Type Free Boundary Problems, Comm. in PDE 29 (2004),1167-1186;
  • Chang, S.-Y. A, and González, M. del Mar : Fractional Laplacian in Conformal Geometry, preprint.
  • Chanillo, S., Grieser, D., Imai, M., Kurata, K., and Ohnishi, I.: Symmetry Breaking and Other Phenomena in theOptimization of eigenvalues for Composite Membranes, Comm. in Math. Physics 214 (2000), 315-337.
  • Chanillo, S., Grieser, D., and Kurata, K.: The Free Boundary in the Optimization of Composite Membranes, ContemporaryMath. of the AMS 268 (2000), 61-81.
  • Chanillo, S., and Kenig, C.: Weak Uniqueness and Partial Regularity in the Composite Membrane problem, J.European Math. Soc. 10 (2008), 705-737.
  • Chanillo, S., Kenig, C., and To, T. : Regularity of the Minimizers in the Composite Membrane Problem in R2, J.Functional Analysis, 255 (2008), 2299-2320.
  • Fradkin, E.S. and Tseytlin, A. A., One loop β-functions in Conformal Supergravities, Nucl. Physics B, 203, (1982),157-178.
  • Fradkin, E.S. and Tseytlin, A. A., Asymptotic Freedom in Extended Conformal Supergravities, Physics Letters B,110B (2), (1982), 117-122.
  • Graham, C. R., Jenne, R., Mason, L. J., and Sparling, G. A. J., : Conformally Invariant Powers of the Laplacian I.Existence, J. London Math. Soc. 46(2), (1992), 557-565.
  • Graham, C. R., and Zworski, M. : Scattering Matrix in Conformal Geometry, Inventiones Math., 152, (2003), 89-118.
  • Lieb, L. and Loss, M. : Analysis, Graduate Studies in Mathematics 14, (1997) American Math. Society, ProvidenceRI.
  • Monneau, R. and Weiss, G. S. : An unstable elliptic free boundary problem arising in solid combustion, Duke Math.J. 136 (2007), 321–341.
  • Shahgholian, H. : The Singular Set for the Composite Membrane problem, Comm. in Math. Physics 217 (2007),93-101.[WoS]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_agms-2012-0002
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