Extremals for Eigenvalues of Laplacians under Conformal Mapping

✍ Scribed by Richard Snyder Laugesen; Carlo Morpurgo

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 441 KB
• Volume: 155
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3222

✦ Synopsis

Po lya and G. Szego showed in 1951 that for simply connected plane domains, the first eigenvalue of the Laplacian (with Dirichlet boundary conditions) is maximal for a disk, under a conformal mapping normalization. That is, if f (z) is a conformal map of a disk D onto a bounded, simply connected plane domain 0, normalized by | f $(0)| =1, then * 1 (0) * 1 (D). Later, Po lya and M. Schiffer showed that actually

, for each n=1, 2, 3,... . This paper shows that for every convex increasing function 8,

, for each n=1, 2, 3,... .

In particular, taking 8(a)=a s for a fixed s>1 gives that the zeta function of the eigenvalues of the Laplacian is minimal for the disk, under Dirichlet boundary conditions. The bulk of the paper addresses similar questions for simply and doubly connected domains on cones and cylinders and on surfaces of variable curvature, extending the work of C. Bandle, T. Gasser, and J. Hersch.

Also, let M g be an N-dim. Riemannian manifold with boundary and for each smooth function w on the closure of M, write * j (w) for the j th eigenvalue of the article no.