Shapes of drums with lowest base frequency under non-isotropic perimeter constraints
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- by Marek Biskup and Evitar B. Procaccia PDF
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Abstract:
We study the minimizers of the sum of the principal Dirichlet eigenvalue of the negative Laplacian and the perimeter with respect to a general norm in the class of Jordan domains in the plane. This is equivalent (modulo scaling) to minimizing the said eigenvalue (or the base frequency of a drum of this shape) subject to a hard constraint on the perimeter. We show that, for all norms, a minimizer exists, is unique up to spatial translations, and is convex but not necessarily smooth. We give conditions on the norm that characterize the appearance of facets and corners. We also demonstrate that near minimizers have to be close to the optimal ones in the Hausdorff distance. Our motivation for considering this class of variational problems comes from a study of random walks in random environment interacting through the boundary of their support.References
- Antonio Auffinger, Michael Damron, and Jack Hanson, 50 years of first-passage percolation, University Lecture Series, vol. 68, American Mathematical Society, Providence, RI, 2017. MR 3729447, DOI 10.1090/ulect/068
- N. Berestycki and A. Yadin, Condensation of random walks and the Wulff crystal, arXiv:1305.0139.
- M. van den Berg and M. Iversen, On the minimization of Dirichlet eigenvalues of the Laplace operator, J. Geom. Anal. 23 (2013), no. 2, 660–676. MR 3023854, DOI 10.1007/s12220-011-9258-0
- Marek Biskup and Eviatar B. Procaccia, Eigenvalue versus perimeter in a shape theorem for self-interacting random walks, Ann. Appl. Probab. 28 (2018), no. 1, 340–377. MR 3770879, DOI 10.1214/17-AAP1307
- Dorin Bucur, Giuseppe Buttazzo, and Antoine Henrot, Minimization of $\lambda _2(\Omega )$ with a perimeter constraint, Indiana Univ. Math. J. 58 (2009), no. 6, 2709–2728. MR 2603765, DOI 10.1512/iumj.2009.58.3768
- Dorin Bucur and Pedro Freitas, Asymptotic behaviour of optimal spectral planar domains with fixed perimeter, J. Math. Phys. 54 (2013), no. 5, 053504, 6. MR 3098942, DOI 10.1063/1.4803140
- R. Cerf, The Wulff crystal in Ising and percolation models, Lecture Notes in Mathematics, vol. 1878, Springer-Verlag, Berlin, 2006. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004; With a foreword by Jean Picard. MR 2241754
- Andrea Colesanti, Brunn-Minkowski inequalities for variational functionals and related problems, Adv. Math. 194 (2005), no. 1, 105–140. MR 2141856, DOI 10.1016/j.aim.2004.06.002
- Guido De Philippis and Bozhidar Velichkov, Existence and regularity of minimizers for some spectral functionals with perimeter constraint, Appl. Math. Optim. 69 (2014), no. 2, 199–231. MR 3175194, DOI 10.1007/s00245-013-9222-4
- R. Dobrushin, R. Kotecký, and S. Shlosman, Wulff construction, Translations of Mathematical Monographs, vol. 104, American Mathematical Society, Providence, RI, 1992. A global shape from local interaction; Translated from the Russian by the authors. MR 1181197, DOI 10.1090/mmono/104
- Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
- Stephen J. Fromm, Potential space estimates for Green potentials in convex domains, Proc. Amer. Math. Soc. 119 (1993), no. 1, 225–233. MR 1156467, DOI 10.1090/S0002-9939-1993-1156467-3
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
- Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682, DOI 10.1007/978-1-4684-9486-0
- Daniel Grieser and David Jerison, The size of the first eigenfunction of a convex planar domain, J. Amer. Math. Soc. 11 (1998), no. 1, 41–72. MR 1470858, DOI 10.1090/S0894-0347-98-00254-9
- Michael Grüter and Kjell-Ove Widman, The Green function for uniformly elliptic equations, Manuscripta Math. 37 (1982), no. 3, 303–342. MR 657523, DOI 10.1007/BF01166225
- Antoine Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. MR 2251558
- Antoine Henrot and Michel Pierre, Variation et optimisation de formes, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 48, Springer, Berlin, 2005 (French). Une analyse géométrique. [A geometric analysis]. MR 2512810, DOI 10.1007/3-540-37689-5
- James Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318. MR 333220, DOI 10.1007/BF00250468
- V. Šverák, On optimal shape design, J. Math. Pures Appl. (9) 72 (1993), no. 6, 537–551. MR 1249408
- Jean E. Taylor, Existence and structure of solutions to a class of nonelliptic variational problems, Symposia Mathematica, Vol. XIV (Convegno di Teoria Geometrica dell’Integrazione e Varietà Minimali, INDAM, Roma, Maggio 1973), Academic Press, London, 1974, pp. 499–508. MR 0420407
- Jean E. Taylor, Unique structure of solutions to a class of nonelliptic variational problems, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 1, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 419–427. MR 0388225
- Bozhidar Velichkov, Existence and regularity results for some shape optimization problems, Tesi. Scuola Normale Superiore di Pisa (Nuova Series) [Theses of Scuola Normale Superiore di Pisa (New Series)], vol. 19, Edizioni della Normale, Pisa, 2015. MR 3309888, DOI 10.1007/978-88-7642-527-1
- Emmanuel Tsukerman and Ellen Veomett, Brunn-Minkowski theory and Cauchy’s surface area formula, Amer. Math. Monthly 124 (2017), no. 10, 922–929. MR 3733300, DOI 10.4169/amer.math.monthly.124.10.922
Additional Information
- Marek Biskup
- Affiliation: Department of Mathematics, University of California Los Angles, Los Angeles, California 90095-1555–and–Center for Theoretical Study, Charles University, Prague, 11000, Czech Republic
- MR Author ID: 630029
- Email: biskup@math.ucla.edu
- Evitar B. Procaccia
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77840
- MR Author ID: 954587
- Email: eviatarp@gmail.com
- Received by editor(s): March 17, 2017
- Received by editor(s) in revised form: January 18, 2018
- Published electronically: April 12, 2019
- Additional Notes: This research was partially supported by NSF grant DMS-1407558 and GAČR project P201/16-15238S
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 71-95
- MSC (2010): Primary 49Q10, 49R05; Secondary 15A42
- DOI: https://doi.org/10.1090/tran/7532
- MathSciNet review: 3968763