On a Pólya functional for rhombi, isosceles triangles, and thinning convex sets. (English) Zbl 1460.52008
The authors state three results. The first one focuses on providing explicitly an upper bound of the product of the first eigenvalue of the Laplacian operator by the norm of the torsion function for an open bounded convex set. The second (resp. third) concentrates on furnishing accurately a lower and an upper bound of Pólya function for an isosceles triangle (resp. a rhombus).
Reviewer: Mohammed El Aïdi (Bogotá)
MSC:
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 47A75 | Eigenvalue problems for linear operators |
| 35J25 | Boundary value problems for second-order elliptic equations |
References:
| [1] | vandenBerg,M.:Estimates for the torsion function and Sobolev constants. Potential Anal.36(2012), no. 4, 607-616. · Zbl 1246.60108 |
| [2] | van den Berg, M.:Spectral bounds for the torsion function.Integral Equations and Operator Theory88(2017), no. 3, 387-400. · Zbl 1378.58024 |
| [3] | van den Berg, M. and Carroll, T.:Hardy inequality andLpestimates for the torsion function.Bull. Lond. Math. Soc.41(2009), no. 6, 980-986. · Zbl 1180.35396 |
| [4] | van den Berg, M., Ferone, V., Nitsch, C. and Trombetti, C.:On P´olya’s inequality for torsional rigidity and first Dirichlet eigenvalue.Integral Equations Operator Theory86(2016), no. 4, 579-600. · Zbl 1388.49012 |
| [5] | Henrot, A.:Extremum problems for eigenvalues of elliptic operators.Frontiers in Mathematics, Birkh¨auser Verlag, Basel, 2006. · Zbl 1109.35081 |
| [6] | Henrot, A., Lucardesi, I. and Philippin, G.:On two functionals involving the maximum of the torsion function.ESAIM Control Optim. Calc. Var.24(2018), no. 4, 1585-1604. · Zbl 1442.35281 |
| [7] | Eggleston, H. G.:Notes on Minkowski geometry. I. Relations between the circumradius, diameter, inradius and minimal width of a convex set.J. London Math. Soc. 33(1958), 76-81. · Zbl 0082.15704 |
| [8] | Payne, L. E.:Bounds for solutions of a class of quasilinear elliptic boundary value problems in terms of the torsion function.Proc. Roy. Soc. Edinburgh Sec. A88 (1981), no. 3-4, 251-265. · Zbl 0481.35022 |
| [9] | P´olya,G. and Szeg¨o,G.:Isoperimetric inequalities in Mathematical Physics. Annals of Mathematics Studies 27, Princeton University Press, Princeton, NJ, 1951. · Zbl 0044.38301 |
| [10] | Qu, C. K. and Wong, R.:“Best possible” upper and lower bounds for the zeroes of the Bessel functionJν(x).Trans. Amer. Math. Soc.351(1999), no. 7, 2833-2859. · Zbl 0930.41020 |
| [11] | Siudeja, B.:Isoperimetric inequalities for eigenvalues of triangles.Indiana Univ. Math. J.59(2010), no. 3, 1097-1120. · Zbl 1220.35105 |
| [12] | Timoshenko, S. P. and Goodier, J. N.:Theory of elasticity.McGraw-Hill Book Company, New York-Toronto-London, 1951. · Zbl 0045.26402 |
| [13] | Vogt,H.:L∞-estimates for the torsion function andL∞-growth of semigroups satisfying Gaussian bounds.Potential Anal.51(2019), no. 1, 37-47. · Zbl 1428.35258 |
| [14] | Yaglom, I. |
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