# zbMATH — the first resource for mathematics

On the lowest eigenvalue of the Laplacian for the intersection of two domains. (English) Zbl 0538.35058
Main result: Let $$A,B\subset {\mathbb{R}}^ n$$ be open and non-empty, and let $$\lambda(A)$$, $$\lambda(B)$$ be the infimum of the $$L^ 2$$-spectrum of $$- \Delta$$ with Dirichlet boundary conditions. Let $$B_ x$$ denote the translate of B by $$x\in {\mathbb{R}}^ n$$. Then $$\inf_{x}\lambda(A\cap B_ x)\leq \lambda(A)+\lambda(B)$$, with strict inequality if A and B are bounded. A similar inequality is proved for $$\lambda_ p(A):= \inf \{\| f\|^ p_ p/\| f\|^ p_ p;$$ $$f\in C_ 0^{\infty}(A),$$ $$f\neq 0\},$$ $$1\leq p<\infty$$. A consequence of this result is a lower bound for $$\sup_{x}vol(A\cap B_ x)$$ in terms of $$\lambda(A)$$, when B is a ball. This result was motivated by the isoperimetric inequality $$\lambda(A)\geq \beta_ n$$ $$vol(A)^{-2/n}$$ where $$\beta_ n$$ is the lowest eigenvalue for a ball of unit volume. A second consequence is a compactness lemma for certain sequences in $$W^{1,p}({\mathbb{R}}^ n)$$, $$1<p<\infty$$.
Reviewer: J.Voigt

##### MSC:
 35P15 Estimates of eigenvalues in context of PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text:
##### References:
  Brascamp, H.J., Lieb, E.H.: Some inequalities for Gaussian measures and the long-range order of the one-dimensional plasma. In: Functional Integration and its Applications, Arthurs, A.M. (ed.) pp. 1-14. Oxford: Clarendon Press 1975 · Zbl 0348.26011  Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis, a Symposium in Honor of Salomon Bochner, Gunning, R.C. (ed.) pp. 145-199. Princeton, N.J.: Princeton University Press 1970 · Zbl 0212.44903  Cheng, S.Y.: On the Hayman-Osserman-Taylor inequality, (preprint).  Croke, C.B.: The first eigenvalue of the Laplacian for plane domains. Proc. Amer. Math. Soc.81, 304-305 (1981) · Zbl 0462.35071  Faber, C.: Beweis das unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitzungsber. Bayer. Akad. der Wiss. Math. Phys., Munich 1923, pp. 169-172 · JFM 49.0342.03  Hayman, W.K.: Some bounds for principle frequency. Applic. Anal.7, 247-254 (1977/1978) · Zbl 0383.35053  Krahn, E.: Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Math. Ann.94, 97-100 (1925) · JFM 51.0356.05  Osserman, R.: A note on Hayman’s theorem on the bass note of a drum. Comment. Math. Helv.52, 545-555 (1977) · Zbl 0374.52008  Osserman, R.: The isoperimetric inequality, Bull. Amer. Math. Soc.84, 1182-1238 (1978) · Zbl 0411.52006  Osserman, R.: Bonnesen-style isoperimetric inequalities. Amer. Math. Monthly86, 1-29 (1979) · Zbl 0404.52012  Taylor, M.: Estimate on the fundamental frequency of a drum. Duke Math. J.46, 447-453 (1979) · Zbl 0418.35068  Yau, S.T.: Isoperimetric constants and the first eigenvalue of a compact manifold. Ann. Sci. Ecolo Norm. Sup.8, 487-507 (1975) · Zbl 0325.53039  Lieb, E.H.: Some vector field equations. In: Proceedings of the March 1983 University of Alabama, Birmingham International Conference on Partial Differential Equations, Knowles, I. (ed.), North-Holland (in press)  Brezis, H., Lieb, E.H.: Minimum action solutions to some vector field equations (in preparation)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.