Optimal convex domains for the first curl eigenvalue in dimension three
HTML articles powered by AMS MathViewer
- by Alberto Enciso, Wadim Gerner and Daniel Peralta-Salas
- Trans. Amer. Math. Soc.
- DOI: https://doi.org/10.1090/tran/8914
- Published electronically: April 24, 2024
- PDF | Request permission
Abstract:
We prove that there exists a bounded convex domain $\Omega \subset \mathbb {R}^3$ of fixed volume that minimizes the first positive curl eigenvalue among all other bounded convex domains of the same volume. We show that this optimal domain cannot be analytic, and that it cannot be stably convex if it is sufficiently smooth (e.g., of class $C^{1,1}$). Existence results for uniformly Hölder optimal domains in a box (that is, contained in a fixed bounded domain $D\subset \mathbb {R}^3$) are also presented.References
- C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998), no. 9, 823–864 (English, with English and French summaries). MR 1626990, DOI 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
- Chérif Amrouche and Nour El Houda Seloula, $L^p$-theory for vector potentials and Sobolev’s inequalities for vector fields: application to the Stokes equations with pressure boundary conditions, Math. Models Methods Appl. Sci. 23 (2013), no. 1, 37–92. MR 2997467, DOI 10.1142/S0218202512500455
- Peter Laurence and Marco Avellaneda, On Woltjer’s variational principle for force-free fields, J. Math. Phys. 32 (1991), no. 5, 1240–1253. MR 1103476, DOI 10.1063/1.529321
- Christian Bär, The curl operator on odd-dimensional manifolds, J. Math. Phys. 60 (2019), no. 3, 031501, 16. MR 3920324, DOI 10.1063/1.5082528
- Dorin Bucur, Regularity of optimal convex shapes, J. Convex Anal. 10 (2003), no. 2, 501–516. MR 2044433
- A. Buffa, M. Costabel, and D. Sheen, On traces for $\textbf {H}(\textbf {curl},\Omega )$ in Lipschitz domains, J. Math. Anal. Appl. 276 (2002), no. 2, 845–867. MR 1944792, DOI 10.1016/S0022-247X(02)00455-9
- Giuseppe Buttazzo and Gianni Dal Maso, An existence result for a class of shape optimization problems, Arch. Rational Mech. Anal. 122 (1993), no. 2, 183–195. MR 1217590, DOI 10.1007/BF00378167
- Jason Cantarella, Dennis DeTurck, Herman Gluck, and Mikhail Teytel, Isoperimetric problems for the helicity of vector fields and the Biot-Savart and curl operators, J. Math. Phys. 41 (2000), no. 8, 5615–5641. MR 1770976, DOI 10.1063/1.533429
- Jason Cantarella, Dennis DeTurck, Herman Gluck, and Mikhail Teytel, The spectrum of the curl operator on spherically symmetric domains, Phys. Plasmas 7 (2000), no. 7, 2766–2775. MR 1766493, DOI 10.1063/1.874127
- Denise Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl. 52 (1975), no. 2, 189–219. MR 385666, DOI 10.1016/0022-247X(75)90091-8
- Alexandre J. Chorin and Jerrold E. Marsden, A mathematical introduction to fluid mechanics, 3rd ed., Texts in Applied Mathematics, vol. 4, Springer-Verlag, New York, 1993. MR 1218879, DOI 10.1007/978-1-4612-0883-9
- Andrea Colesanti and Michele Fimiani, The Minkowski problem for torsional rigidity, Indiana Univ. Math. J. 59 (2010), no. 3, 1013–1039. MR 2779070, DOI 10.1512/iumj.2010.59.3937
- S. Dekel and D. Leviatan, Whitney estimates for convex domains with applications to multivariate piecewise polynomial approximation, Found. Comput. Math. 4 (2004), no. 4, 345–368. MR 2097212, DOI 10.1007/s10208-004-0096-3
- M. C. Delfour and J.-P. Zolésio, Shapes and geometries, Advances in Design and Control, vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. Analysis, differential calculus, and optimization. MR 1855817
- Alberto Enciso, M. Ángeles García-Ferrero, and Daniel Peralta-Salas, The Biot-Savart operator of a bounded domain, J. Math. Pures Appl. (9) 119 (2018), 85–113 (English, with English and French summaries). MR 3862144, DOI 10.1016/j.matpur.2017.11.004
- A. Enciso, A. Luque, and D. Peralta-Salas, MHD equilibria with nonconstant pressure in nondegenerate toroidal domains, J. Eur. Math. Soc. in press (2023).
- Alberto Enciso and Daniel Peralta-Salas, Submanifolds that are level sets of solutions to a second-order elliptic PDE, Adv. Math. 249 (2013), 204–249. MR 3116571, DOI 10.1016/j.aim.2013.08.026
- A. Enciso and D. Peralta-Salas, Non-existence of axisymmetric optimal domains with smooth boundary for the first curl eigenvalue, Ann. Sc. Norm. Sup. Pisa 24 (2023) 311–327.
- N. D. Filonov, The operator $\rm rot$ in domains of finite measure, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 262 (1999), no. Issled. po Lineĭn. Oper. i Teor. Funkts. 27, 227–230, 236 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 110 (2002), no. 5, 3029–3030. MR 1734339, DOI 10.1023/A:1015303807742
- Wadim Gerner, Existence and characterisation of magnetic energy minimisers on oriented, compact Riemannian 3-manifolds with boundary in arbitrary helicity classes, Ann. Global Anal. Geom. 58 (2020), no. 3, 267–285. MR 4145737, DOI 10.1007/s10455-020-09727-4
- W. Gerner, Minimisation Problems in Ideal Magnetohydrodynamics, PhD dissertation, RWTH Aachen University, 2020.
- Antoine Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. MR 2251558
- Antoine Henrot (ed.), Shape optimization and spectral theory, De Gruyter Open, Warsaw, 2017. MR 3681143
- Antoine Henrot and Edouard Oudet, Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions, Arch. Ration. Mech. Anal. 169 (2003), no. 1, 73–87. MR 1996269, DOI 10.1007/s00205-003-0259-4
- Ralf Hiptmair, Peter Robert Kotiuga, and Sébastien Tordeux, Self-adjoint curl operators, Ann. Mat. Pura Appl. (4) 191 (2012), no. 3, 431–457. MR 2958342, DOI 10.1007/s10231-011-0189-y
- D. Hug and W. Weil, Lectures on Convex Geometry, Springer, Switzerland, 2020.
- Fritz John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, New York, 1948, pp. 187–204. MR 30135
- Dennis Kriventsov and Fanghua Lin, Regularity for shape optimizers: the nondegenerate case, Comm. Pure Appl. Math. 71 (2018), no. 8, 1535–1596. MR 3847749, DOI 10.1002/cpa.21743
- Steven G. Krantz, Convex analysis, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015. MR 3243119
- Dorina Mitrea, Marius Mitrea, and Michael Taylor, Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds, Mem. Amer. Math. Soc. 150 (2001), no. 713, x+120. MR 1809655, DOI 10.1090/memo/0713
- Piotr B. Mucha and Milan Pokorný, The rot-div system in exterior domains, J. Math. Fluid Mech. 16 (2014), no. 4, 701–720. MR 3267543, DOI 10.1007/s00021-014-0181-6
- Jun O’Hara, Minimal unfolded regions of a convex hull and parallel bodies, Hokkaido Math. J. 44 (2015), no. 2, 175–183. MR 3532105, DOI 10.14492/hokmj/1470053289
- Rainer Picard, On a selfadjoint realization of curl and some of its applications, Ricerche Mat. 47 (1998), no. 1, 153–180. MR 1760328
- Gregory Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572–611. MR 769382, DOI 10.1016/0022-1236(84)90066-1
- Jean-Philippe Vial, Strong convexity of sets and functions, J. Math. Econom. 9 (1982), no. 1-2, 187–205. MR 637263, DOI 10.1016/0304-4068(82)90026-X
- M. D. Wills, Hausdorff distance and convex sets, J. Convex Anal. 14 (2007), no. 1, 109–117. MR 2310432
- Zensho Yoshida and Yoshikazu Giga, Remarks on spectra of operator rot, Math. Z. 204 (1990), no. 2, 235–245. MR 1055988, DOI 10.1007/BF02570870
Bibliographic Information
- Alberto Enciso
- Affiliation: Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain
- MR Author ID: 751606
- Email: aenciso@icmat.es
- Wadim Gerner
- Affiliation: Sorbonne Université, Inria, CNRS, Laboratoire Jacques-Louis Lions (LJLL), Paris, France
- MR Author ID: 1399723
- ORCID: 0000-0002-0483-178X
- Email: wadim.gerner@icmat.es
- Daniel Peralta-Salas
- Affiliation: Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain
- MR Author ID: 648494
- Email: dperalta@icmat.es
- Received by editor(s): May 23, 2022
- Received by editor(s) in revised form: January 16, 2023
- Published electronically: April 24, 2024
- Additional Notes: This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme through the grant agreement 862342 (A.E.). It is partially supported by the grants CEX2019-000904-S and PID2019-106715GB GB-C21 (D.P.-S.) funded by MCIN/AEI/10.13039/501100011033.
- © Copyright 2024 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
- MSC (2020): Primary 35P05, 52A15, 49Q10
- DOI: https://doi.org/10.1090/tran/8914