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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Windows of given area
with minimal heat diffusion

Author: Jochen Denzler
Journal: Trans. Amer. Math. Soc. 351 (1999), 569-580
MSC (1991): Primary 49J40; Secondary 49J10, 35J20, 35R05
MathSciNet review: 1475680
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Abstract | References | Similar Articles | Additional Information

Abstract: For a bounded Lipschitz domain $\Omega$, we show the existence of a measurable set $D\subset \partial\Omega$ of given area such that the first eigenvalue of the Laplacian with Dirichlet conditions on $D$ and Neumann conditions on $\partial \Omega \setminus D$ becomes minimal. If $\Omega$ is a ball, $D$ will be a spherical cap.

References [Enhancements On Off] (What's this?)

  • 1. R.A. Adams: Sobolev Spaces, Academic Press, 1978 MR 56:9247
  • 2. S. Agmon, A. Douglis, L. Nirenberg: Estimates near the Boundary for Solutions of Elliptic Partial Differential Equations Satisfying General Boundary Conditions, Comm. Pure Appl. Math. 12 (1959), 623-727 MR 23:A2610
  • 3. H.J. Brascamp, E.H. Lieb, J.M. Luttinger: A general rearrangement inequality for multiple integrals, Journal of Functional Analysis 17 (1974), 227-237 MR 49:10835
  • 4. J. Denzler: Bounds for the heat diffusion through windows of given area, J. Math. Anal. Appl. 217 (1998), 405-422. CMP 98:07
  • 5. H. Federer: Geometric Measure Theory, Springer 1969 (Grundlehren 153, now reprinted in the series Classics in Mathematics) MR 41:1976
  • 6. D. Gilbarg, N.S. Trudinger: Elliptic Partial Differential Equations of Second Order, Springer Grundlehren 224 MR 57:13109
  • 7. D.S. Jerison, C.E. Kenig: The Neumann problem on Lipschitz domains, Bull. AMS 4 (1981), 203-207 MR 84a:35064
  • 8. C.E. Kenig: Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS, Vol. 83, American Mathematical Society, 1994 MR 96a:35040
  • 9. C.E. Kenig, J. Pipher: The Neumann problem for elliptic equations with non-smooth coefficients, Inventiones mathematicae 113 (1993), 447-509 MR 95b:35046
  • 10. O.A. Ladyzhenskaya, N.N. Ural'tseva: Linear and Quasilinear Elliptic Equations, Academic Press 1968 MR 39:5941
  • 11. G. Pólya, G. Szego: Isoperimetrical Inequalities in Mathematical Physics, Annals of Mathematics Studies 27, Princeton Univ. Press, 1951 MR 13:270d
  • 12. F. Riesz: Sur une inégalité integrale, Journal of the London Mathematical Society 5 (1930), 162-168
  • 13. E. Schmidt: Die Brunn-Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und nichteuklidischen Geometrie; part I: Mathematische Nachrichten 1 (1948), 81-157; part II: 2 (1949), 171-244. MR 10:471d; MR 11:534l
  • 14. S. Sobolev: Ob odnoi teoreme funktsional'nogo analiza (On a theorem of functional analysis), Matematicheskii Sbornik N.S. 4(46) (1938), 471-497
  • 15. E. Sperner: Zur Symmetrisierung von Funktionen auf Sphären, Mathematische Zeitschrift 134 (1973), 317-327 MR 49:5310
  • 16. E. Sperner: Spherical Symmetrization and Eigenvalue estimates, Mathematische Zeitschrift 176 (1981), 75-86 MR 82e:35062
  • 17. G. Stampacchia: Problemi al contorno ellittici, con dati discontinui, dotati di soluzioni hölderiane, Annali di Matematica Pura ed Applicata 51 (1960), 1-37 MR 23:A3897

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Additional Information

Jochen Denzler
Affiliation: Mathematisches Institut, Ludwig–Maximilians–Universität, Theresienstraße 39, D–80333 München, Germany
Address at time of publication: Zentrum Mathematik, Technische Universität, Arcisstrasse 21, D-80290 München, Germany

Received by editor(s): November 16, 1996
Article copyright: © Copyright 1999 American Mathematical Society

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