Conformal Geometry and the Composite Membrane Problem

Sagun Chanillo

Analysis and Geometry in Metric Spaces (2013)

  • Volume: 1, page 31-35
  • ISSN: 2299-3274

Abstract

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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.

How to cite

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Sagun Chanillo. "Conformal Geometry and the Composite Membrane Problem." Analysis and Geometry in Metric Spaces 1 (2013): 31-35. <http://eudml.org/doc/266809>.

@article{SagunChanillo2013,
abstract = {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.},
author = {Sagun Chanillo},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {Eigenvalue Minimization in Conformal classes; GJMS operators; Composite Membrane problem; Free Boundary Problems; Conformal Geometry; Paneitz operator; eigenvalue minimization in conformal classes; composite membrane problem; free boundary problems; conformal geometry; paneitz operator},
language = {eng},
pages = {31-35},
title = {Conformal Geometry and the Composite Membrane Problem},
url = {http://eudml.org/doc/266809},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Sagun Chanillo
TI - Conformal Geometry and the Composite Membrane Problem
JO - Analysis and Geometry in Metric Spaces
PY - 2013
VL - 1
SP - 31
EP - 35
AB - 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.
LA - eng
KW - Eigenvalue Minimization in Conformal classes; GJMS operators; Composite Membrane problem; Free Boundary Problems; Conformal Geometry; Paneitz operator; eigenvalue minimization in conformal classes; composite membrane problem; free boundary problems; conformal geometry; paneitz operator
UR - http://eudml.org/doc/266809
ER -

References

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  2. Blank, Ivan: Eliminating Mixed Asymptotics in Obstacle Type Free Boundary Problems, Comm. in PDE 29 (2004), 1167-1186; Zbl1082.35165
  3. Chang, S.-Y. A, and González, M. del Mar : Fractional Laplacian in Conformal Geometry, preprint. 
  4. Chanillo, S., Grieser, D., Imai, M., Kurata, K., and Ohnishi, I.: Symmetry Breaking and Other Phenomena in the Optimization of eigenvalues for Composite Membranes, Comm. in Math. Physics 214 (2000), 315-337. Zbl0972.49030
  5. Chanillo, S., Grieser, D., and Kurata, K.: The Free Boundary in the Optimization of Composite Membranes, Contemporary Math. of the AMS 268 (2000), 61-81. Zbl0988.35124
  6. Chanillo, S., and Kenig, C.: Weak Uniqueness and Partial Regularity in the Composite Membrane problem, J. European Math. Soc. 10 (2008), 705-737. Zbl1154.35096
  7. 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. Zbl1154.49026
  8. Fradkin, E.S. and Tseytlin, A. A., One loop β-functions in Conformal Supergravities, Nucl. Physics B, 203, (1982), 157-178. 
  9. Fradkin, E.S. and Tseytlin, A. A., Asymptotic Freedom in Extended Conformal Supergravities, Physics Letters B, 110B (2), (1982), 117-122. 
  10. 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. Zbl0726.53010
  11. Graham, C. R., and Zworski, M. : Scattering Matrix in Conformal Geometry, Inventiones Math., 152, (2003), 89-118. Zbl1030.58022
  12. Lieb, L. and Loss, M. : Analysis, Graduate Studies in Mathematics 14, (1997) American Math. Society, Providence RI. 
  13. Monneau, R. and Weiss, G. S. : An unstable elliptic free boundary problem arising in solid combustion, Duke Math. J. 136 (2007), 321–341. Zbl1119.35123
  14. Shahgholian, H. : The Singular Set for the Composite Membrane problem, Comm. in Math. Physics 217 (2007), 93-101. [WoS] Zbl1157.35125

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