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Blaschke-Santaló and Mahler inequalities for the first eigenvalue of the Dirichlet Laplacian. (English) Zbl 1361.49019

The \(\lambda_1\)-product functional associates with a given convex body \(K\subset {\mathbb R}^n\) the quantity \(\lambda_1(K)\lambda_1(K^o)\), where \(K^o\) is the polar body of \(K\) and \(\lambda_1(\cdot)\) is the first eigenvalue of the Dirichlet Laplacian.
The first result in this paper, Theorem 1, shows that the minimum of the \(\lambda_1\)-product functional in the class of centrally symmetric convex bodies is attained by balls: when \(K\) is centrally symmetric and \(B\) is a ball, there holds \[ \lambda_1(K)\lambda_1(K^o)\geq \lambda_1(B)\lambda_1(B^o). \] The proof of this inequality follows from the classical Blaschke-Santaló’s and Faber-Krahn inequalities.
The second main result is Theorem 9, where it is shown that the supremum of the planar functional \[ \inf_{T\in D_2} \lambda_1(T(K))\lambda_1(T(K)^o) \] in \({\mathcal K}_{\#}^2\) is achieved by the square. Here \({\mathcal K}_{\#}^2\) is the set of unconditional bodies (i.e. those symmetric with respect to all coordinate hyperplanes of a fixed frame) and \(D_2\) is the class of invertible diagonal transformations of \({\mathbb R}^2\). As the authors show in Remark 3, the restriction on the space is necessary since the supremum of the \(\lambda_1\)-product functional is \(+\infty\) in the class of convex bodies and in the one of centrally symmetric convex bodies.
For the proof of Theorem 9, the authors first show that a sufficient condition for the square to be a minimizer is that it solves the problem \[ \sup\{[\lambda_1(\Omega)|\Omega|]:\Omega\in {\mathcal O}\}, \] where \({\mathcal O}\) is the class of convex axisymmetric octagons having their vertices lying on the axes at the same distance. The proof of the latter result, Theorem 12, is given by a hybrid method involving both theoretical and numerical tools.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A40 Inequalities and extremum problems involving convexity in convex geometry
49R05 Variational methods for eigenvalues of operators
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[1] Alt, Existence and regularity for a minimum problem with free boundary, J. reine angew. Math. 325 pp 105– (1981) · Zbl 0449.35105
[2] DOI: 10.1007/s10957-011-9983-3 · Zbl 1252.90076 · doi:10.1007/s10957-011-9983-3
[3] DOI: 10.2140/pjm.2004.217.201 · Zbl 1078.35080 · doi:10.2140/pjm.2004.217.201
[4] Ball, Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. (2) 44 pp 351– (1991) · Zbl 0694.46010 · doi:10.1112/jlms/s2-44.2.351
[5] K. Ball , An elementary introduction to modern convex geometry, Flavors of Geometry, Mathematical Sciences Research Institute Publications 31 (Cambridge University Press, Cambridge, 1997) 1–58. · Zbl 0901.52002
[6] DOI: 10.1007/s00605-013-0499-9 · Zbl 1288.52003 · doi:10.1007/s00605-013-0499-9
[7] W. Blaschke , Vorlesungen über Integralgeometrie, 3te Aufl (Deutscher Verlag der Wissenschaften, Berlin, 1955). · Zbl 0066.40703
[8] DOI: 10.1007/BF01456879 · Zbl 0546.31001 · doi:10.1007/BF01456879
[9] DOI: 10.1007/BF01455935 · Zbl 0584.31003 · doi:10.1007/BF01455935
[10] DOI: 10.1016/j.aim.2010.04.014 · Zbl 1216.52007 · doi:10.1016/j.aim.2010.04.014
[11] Böröczky, On the volume product of planar polar convex bodies–lower estimates with stability, Studia Sci. Math. Hungar. 50 pp 159– (2013)
[12] DOI: 10.1007/BF01388911 · Zbl 0617.52006 · doi:10.1007/BF01388911
[13] DOI: 10.1002/mana.19951720104 · Zbl 0886.49010 · doi:10.1002/mana.19951720104
[14] DOI: 10.1007/BF02829490 · Zbl 0965.49002 · doi:10.1007/BF02829490
[15] DOI: 10.1007/s00205-012-0561-0 · Zbl 1254.35165 · doi:10.1007/s00205-012-0561-0
[16] Colbois, Isoperimetric control of the spectrum of a compact hypersurface, J. reine angew. Math. 683 pp 49– (2013)
[17] DOI: 10.1016/j.aim.2004.06.002 · Zbl 1128.35318 · doi:10.1016/j.aim.2004.06.002
[18] DOI: 10.1512/iumj.2010.59.3937 · Zbl 1217.31001 · doi:10.1512/iumj.2010.59.3937
[19] El Soufi, Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds, Trans. Amer. Math. Soc. 361 pp 2337– (2009) · Zbl 1162.58009 · doi:10.1090/S0002-9947-08-04780-6
[20] DOI: 10.1007/s00209-006-0078-z · Zbl 1128.52007 · doi:10.1007/s00209-006-0078-z
[21] DOI: 10.1016/j.aim.2008.03.013 · Zbl 1153.52003 · doi:10.1016/j.aim.2008.03.013
[22] DOI: 10.4310/CAG.2004.v12.n5.a5 · Zbl 1110.58003 · doi:10.4310/CAG.2004.v12.n5.a5
[23] DOI: 10.1090/S0002-9939-08-09399-4 · Zbl 1147.58030 · doi:10.1090/S0002-9939-08-09399-4
[24] DOI: 10.1051/cocv/2009018 · Zbl 1205.35174 · doi:10.1051/cocv/2009018
[25] E. Harrell , A. Henrot and J. Lamboley , ’On the local minimizers of the Mahler volume’, Preprint, 2014, http://arxiv.org/abs/1104.3663 . · Zbl 1330.49043
[26] A. Henrot , Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics (Birkhäuser, Basel, 2006). · Zbl 1109.35081
[27] DOI: 10.1007/BF02547334 · Zbl 0880.35041 · doi:10.1007/BF02547334
[28] DOI: 10.1006/aima.1996.0062 · Zbl 0920.35056 · doi:10.1006/aima.1996.0062
[29] DOI: 10.1112/S0025579310001555 · Zbl 1226.52002 · doi:10.1112/S0025579310001555
[30] DOI: 10.1007/s00039-008-0669-4 · Zbl 1169.52004 · doi:10.1007/s00039-008-0669-4
[31] DOI: 10.4310/CAG.2011.v19.n5.a2 · Zbl 1253.35087 · doi:10.4310/CAG.2011.v19.n5.a2
[32] DOI: 10.1007/PL00000076 · Zbl 0916.52003 · doi:10.1007/PL00000076
[33] Lutwak, Blaschke–Santaló inequalities, J. Differ. Geom. 47 pp 1– (1997) · Zbl 0906.52003 · doi:10.4310/jdg/1214460036
[34] Mahler, Ein Übertragungsprinzip für konvexe Körper, Časopis Pěst. Mat. Fys. 68 pp 93– (1939)
[35] MATLAB and Partial Differential Equation Toolbox Release 2016a, The MathWorks, Inc., Natick, MA, USA.
[36] DOI: 10.1016/j.matpur.2013.01.008 · Zbl 1296.35100 · doi:10.1016/j.matpur.2013.01.008
[37] DOI: 10.1007/BF02765029 · Zbl 0629.46023 · doi:10.1007/BF02765029
[38] DOI: 10.1007/BF01199119 · Zbl 0718.52011 · doi:10.1007/BF01199119
[39] DOI: 10.1215/00127094-2010-042 · Zbl 1207.52005 · doi:10.1215/00127094-2010-042
[40] DOI: 10.1051/cocv:2004011 · Zbl 1076.74045 · doi:10.1051/cocv:2004011
[41] DOI: 10.1007/BF01164009 · Zbl 0578.52005 · doi:10.1007/BF01164009
[42] DOI: 10.1007/s10711-013-9917-3 · Zbl 1320.52010 · doi:10.1007/s10711-013-9917-3
[43] J. Saint-Raymond , ’Sur le volume des corps convexes symétriques’, Initiation seminar on analysis: G. Choquet-M. Rogalski-J. Saint-Raymond, 20th Year: 1980/1981, Publications mathématiques de l’Université Pierre et Marie Curie 46 (1980), Exp. No. 11, 25.
[44] Santaló, An affine invariant for convex bodies of \(n\) -dimensional space, Port. Math. 8 pp 155– (1949)
[45] R. Schneider , Convex bodies: the Brunn–Minkowski theory, Expanded edn, Encyclopedia of Mathematics and its Applications 151 (Cambridge University Press, Cambridge, 2014).
[46] T. Tao , Structure and randomness, American Mathematical Society (Providence, RI, 2008). Pages from year one of a mathematical blog. · Zbl 1245.00024
[47] DOI: 10.1112/blms/bdu106 · Zbl 1317.49052 · doi:10.1112/blms/bdu106
[48] DOI: 10.1007/s12220-011-9258-0 · Zbl 1262.49044 · doi:10.1007/s12220-011-9258-0
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