Abstract
Let \(\Omega \) be an open set in Euclidean space with finite Lebesgue measure \(\vert \Omega \vert \). We obtain some properties of the set function \(F:\Omega \mapsto {\mathbb {R}}^+\) defined by
where \(T(\Omega )\) and \(\lambda _1(\Omega )\) are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical Pólya bound \(F(\Omega )\le 1,\) and show that
where \(\nu _m\) depends only on m. For any \(m=2,3,\ldots \) and \(\epsilon \in (0,1)\) we construct an open set \(\Omega _{\epsilon }\subset {\mathbb {R}}^m\) such that \(F(\Omega _{\epsilon })\ge 1-\epsilon \).
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References
Abramowitz, M., Stegun, I.A.: Pocketbook of Mathematical Functions. Verlag Harri Deutsch, Thun (1984)
Bandle, C.: Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics. Pitman, London (1980)
van den Berg, M.: Large time asymptotics of the heat flow. Quart. J. Math. Oxf. Ser. 41, 245–253 (1990)
van den Berg, M., Buttazzo, G., Velichkov, B.: Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity. In: Pratelli, A., Leugering, G. (eds.) New Trends in Shape Optimization, Int. Series Numerical Math, vol. 166, pp. 19–41. Birkhäuser, Basel (2016)
van den Berg, M., Davies, E.B.: Heat flow out of regions in \({\mathbb{R}}^m\). Math. Z. 202, 463–482 (1989)
Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems, Progress in Nonlinear Differential Equations and their Applications 65. Birkhäuser Boston Inc, Boston (2005)
Della Pietra, F., Gavitone, N.: Sharp bounds for the first eigenvalue and the torsional rigidity related to some anisotropic operators. Math. Nachr. 287, 194–209 (2014)
Henrot, A.: Extremum problems for eigenvalues of elliptic operators. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006)
John, F.: Extremum problems with inequalities as subsidiary conditions. Studies and Essays Presented to R. Courant on his 60’th Birthday, January 8, 1948, pp. 187–204. Interscience Publishers, Inc., New York, NY (1948)
Makai, E.: On the principal frequency of a membrane and the torsional rigidity of a beam. Studies in Mathematical Analysis and Realted Topics: Essays in Honor of George Pólya, pp. 227–231. Stanford University Press, Stanford (1962)
Markvorsen, S., Palmer, V.: Torsional rigidity of minimal submanifolds. Proc. Lond. Math. Soc. 93, 253–272 (2006)
Maz’ya, V., Nazarov, S.A., Plamenevskiĭ, B.A.: Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Birkhäuser Verlag, Basel (2000)
Port, S.C., Stone, C.J.: Brownian Motion and Classical Potential Theory. Academic Press, New York (1967)
Pólya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics, Ann. of Math. Stud. 27. Princeton University Press, Princeton (1951)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV, Analysis of Operators. Academic Press, New York (1978)
Talenti, G.: Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa 3, 697–718 (1976)
Taylor, M.E.: Scattering length and perturbations of \(-\Delta \) by positive potentials. J. Math. Anal. Appl. 53, 291–312 (1976)
Taylor, M.E.: Estimate on the fundamental frequency of a drum. Duke Math. J. 46, 447–453 (1979)
Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity. McGraw-Hill Book Company Inc, New York (1951)
Yaglom, I.M., Boltyanskii, V.G.: Convex figures, (Translated by P. J. Kelly, L. F. Walton), Holt, Rinehart and Winston, New York (1961)
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MvdB acknowledges support by The Leverhulme Trust through International Network Grant Laplacians, Random Walks, Bose Gas, Quantum Spin Systems.
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van den Berg, M., Ferone, V., Nitsch, C. et al. On Pólya’s Inequality for Torsional Rigidity and First Dirichlet Eigenvalue. Integr. Equ. Oper. Theory 86, 579–600 (2016). https://doi.org/10.1007/s00020-016-2334-x
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DOI: https://doi.org/10.1007/s00020-016-2334-x