Windows of given area with minimal heat diffusion

Denzler, Jochen

doi:10.1090/S0002-9947-99-02207-2

Windows of given area with minimal heat diffusion
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by Jochen Denzler

Trans. Amer. Math. Soc. 351 (1999), 569-580

DOI: https://doi.org/10.1090/S0002-9947-99-02207-2

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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

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI 10.1002/cpa.3160120405
H. J. Brascamp, Elliott H. Lieb, and J. M. Luttinger, A general rearrangement inequality for multiple integrals, J. Functional Analysis 17 (1974), 227–237. MR 0346109, DOI 10.1016/0022-1236(74)90013-5
J. Denzler: Bounds for the heat diffusion through windows of given area, J. Math. Anal. Appl. 217 (1998), 405–422.
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443
David S. Jerison and Carlos E. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 203–207. MR 598688, DOI 10.1090/S0273-0979-1981-14884-9
Carlos E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, vol. 83, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1282720, DOI 10.1090/cbms/083
Carlos E. Kenig and Jill Pipher, The Neumann problem for elliptic equations with nonsmooth coefficients, Invent. Math. 113 (1993), no. 3, 447–509. MR 1231834, DOI 10.1007/BF01244315
Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
F. Riesz: Sur une inégalité integrale, Journal of the London Mathematical Society 5 (1930), 162–168
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
S. Sobolev: Ob odnoĭ teoreme funktsional’nogo analiza (On a theorem of functional analysis), Matematicheskiĭ Sbornik N.S. 4(46) (1938), 471–497
Emanuel Sperner Jr., Zur Symmetrisierung von Funktionen auf Sphären, Math. Z. 134 (1973), 317–327 (German). MR 340558, DOI 10.1007/BF01214695
Emanuel Sperner Jr., Spherical symmetrization and eigenvalue estimates, Math. Z. 176 (1981), no. 1, 75–86. MR 606173, DOI 10.1007/BF01258906
Guido Stampacchia, Problemi al contorno ellitici, con dati discontinui, dotati di soluzionie hölderiane, Ann. Mat. Pura Appl. (4) 51 (1960), 1–37 (Italian). MR 126601, DOI 10.1007/BF02410941