Isoperimetric inequalities for eigenvalues of the Laplacian and the Schrödinger operator (Q2908962)

This paper is an extended version of lectures given by the first author in connection with the International Conference on the isoperimetric problem of Queen Dido and its mathematical ramifications.NEWLINENEWLINEThe following topics are presentedNEWLINENEWLINE1. IntroductionNEWLINENEWLINE2. Can one hear the shape of the drum?NEWLINENEWLINE3. RearrangementsNEWLINENEWLINE4. The Rayleigh-Faber-Krahn inequalityNEWLINENEWLINE5. The Szegö-Weinberger inequalityNEWLINENEWLINE6. The Payne-Pólya-Weinberger inequalityNEWLINENEWLINE7. The fundamental gapNEWLINENEWLINE8. An isoperimetric inequality for ovals in the planeNEWLINENEWLINE9. Fourth order differential operator.NEWLINENEWLINEAfter a short introduction to isoperimetric problems with some historical remarks, Kac's famous problem is discussed including recent developments. In chapter 3 the definition and some fundamental facts for rearrangements are given. The famous Rayleigh-Faber-Krahn inequality is the topic of chapter 4. It follows a sketch of the proof for a domain \(\Omega \subset R^n \) with rearrangements techniques. The extension of the Rayleigh-Faber-Krahn inequality to Schrödinger operators, the Gaussian space (that means \(R^n\) endowed with the measure \(d\mu = \gamma (x)d^n x, \;\gamma (x) = (2\pi)^{-n/2}e^{-|x|^2/2}\)), and spaces of constant non-zero curvature are also given in this chapter. At the end of chapter 4 the Robin boundary value problem is mentioned and the proof of the Rayleigh-Faber-Krahn inequality for the Robin problem by Daners is given, see also the recent article by \textit{D. Daners} [Arch. Math. 96, No. 2, 187--199 (2011; Zbl 1210.35250)].NEWLINENEWLINEThe next chapter deals with the analogy to the Rayleigh-Faber-Krahn inequality for the Laplace operator with Neumann boundary conditions, i.e. the Szegö-Weinberger inequality, which states that, among all bounded domains in \(R^n\) with the same volume, the ball maximizes the first nontrivial eigenvalue. It is well-known [\textit{C. Bandle}, Isoperimetric Inequalities and Applications. Monographs and Studies in Mathematics, 7. Boston, London, Melbourne: Pitman Advanced Publishing Program (1980; Zbl 0436.35063); \textit{A. Henrot}, Extremum problems for eigenvalues of elliptic operators. Frontiers in Mathematics. Basel: Birkhäuser (2006; Zbl 1109.35081)] that the eigenvalue problem of the Laplacian with free boundary conditions is much more difficult than the eigenvalue problem with fixed boundary conditions. For the eigenvalues of the Dirichlet-Laplacian on bounded domains, it was conjectured in 1955 by \textit{L. E. Payne, G. Pólya} and \textit{H. F. Weinberger} [C. R. Acad. Sci., Paris 241, 917--919 (1955; Zbl 0065.08801)] that the ratio \(\lambda_2(\Omega)/\lambda_1(\Omega), \Omega \subset R^2\) is maximized for a disk. This Payne-Pólya-Weinberger inequality can be stated in the following form: \(\lambda_2(\Omega) \leq \lambda_2(S_1)\), \(\Omega \subset R^n\), with \(S_1 \subset R^n\) a ball with \(\lambda_1(\Omega) = \lambda_1(S_1) \), with equality if and only if \(\Omega\) is a ball. A proof was given by \textit{M. S. Ashbaugh} and \textit{R. D. Benguria} [Bull. Am. Math. Soc., New Ser. 25, No. 1, 19--29 (1991; Zbl 0736.35075); Ann. Math. (2) 135, No. 3, 601--628 (1992; Zbl 0757.35052); Commun. Math. Phys. 147, No. 1, 181--190 (1992; Zbl 0758.34075)]. The full proof is given in the lectures.NEWLINENEWLINETwo generalizations of the Payne-Pólya-Weinberger inequality are mentioned. The first has to do with the Schrödinger operator and was given by \textit{R. D. Benguria} and \textit{H. Linde} [Commun. Math. Phys. 267, No. 3, 741--755 (2006; Zbl 1126.35035)] and the second with spaces of constant curvature by \textit{M. S. Ashbaugh} and \textit{R. D. Benguria} [Trans. Am. Math. Soc. 353, No. 3, 1055--1087 (2001; Zbl 0981.53023)].NEWLINENEWLINEThe spectral gap means \(\lambda_2 - \lambda_1 \) for the Dirichlet eigenvalues of the Laplacian. Here, some results for Schrödinger operators of the form \(H = -\Delta + V \) on compact, convex domains in \(R^n\), with Dirichlet boundary conditions and semiconvex potentials have been considered, especially the sharp lower bound proven by \textit{B. Andrews} and \textit{J. Clutterbuck} [J. Am. Math. Soc. 24, No. 3, 899--916 (2011; Zbl 1222.35130)]. Chapter 8 deals with a conjectured isoperimetric inequality for closed, smooth curves in the plane, which has connections to a special case of the Lieb-Thirring inequality. Let \(C\) be a closed curve in the plane with positive curvature \(\kappa \) and let \(H(C) \equiv -\frac{d^2}{ds^2}+\kappa^2\) and \(\lambda_1(C)\) denote the lowest eigenvalue of \( H(C) \); then the conjecture is \(\lambda_1(C) \geq 1\). The conjecture is still open. In the last chapter a short discussion of the vibrations of a clamped plate, the buckling problem and the vibrations of the free vibrating plate are presented.NEWLINENEWLINEBibliographical notes complete every chapter. These high-quality lectures are an essential source for researchers in this field.