Kac' problem: Which kind of domains can one know those shapes by the eigenvalues of Laplacian with Dirichlet data?
speaker: Waichiro Matsumoto (Ryukoku University)abstract: CAN ONE HEAR THE SHAPE OF A NONCONVEX DRUM? W. MATSUMOTO, M. MURAI AND S. YOTSUTANI M. Kac 3 asked \Can one hear the shape of a drum? " and pointed out that if one hear the same sounds as those of a sphere, the drum must be the same sphere. (We say that on a sphere, the uniqueness holds.) S. Marvizi and R. Melrose 4 and K. Watanabe 5 and 6 gave di erent families of smooth convex domains with the uniqueness. Either of them is obtained through the variational problem on a coeÆcient of the asymptotic expansion of the trace of the fundamental solution to the wave equation or the heat equation with the Dirichlet condition. S. Zelditch 7 gave a suÆcient condition for the uniqueness. On the other hand, C. Gordon, D. Webb and S. Wolpert 2 gave a counter example, which has been simpli ed by S. J. Chapman 1. Their counter examples are nonconvex polygons. There rest the questions; (1) Does there exist a nonconvex domain on which the answer is yes? (2) Does there exist a convex counter example? (3) Does there exist a smooth counter example? (4) What is the necessary and suÆcient condition for the uniqueness? In these questions, the rst one is rather easy to answer, because K. Watanabe sug- gested that his domains may include nonconvex ones by the numerical try. In this lectures, we will give a rigorous proof and show the shapes of nonconvex drums with the unique- ness. We call the curves determined by the Euler equation of Watanabe's variational problem \the minimal curvature energy curve". The Euler equation is nonlinear and its coeÆ- cients include the integrals of powers of the curvature and of the square of the deirvative of the curvature. This make it diÆcult to solve the equaqtion. How to solve the equations of this type, it itself is an interest. The global structure of the minimal curvature energy curves is also interesting on the view point of the classical di erential geometry. References 1 S. J. Chapman; Drums that sound the same, Amer. Math. Mon. 102 (1995), 124-138. 2 C. Gordon, D. Webb and S. Wolpert; Isospectral plane domains and surfaces via Riemannian orb- ifolds, Inv. Math. 110 (1992), 1-22. 3 M. Kac; Can one hear the shape of a dram?, Amer. Math. Mon. 73 (1966), 1-23. 4 S. Marvizi and R. Melrose; Spectral invariants of convex planar regions, Jour. Di . Geom. 17 (1982), 475-502 5 K. Watanabe; Plane domains which are spectrally determined, Ann. Global Anal. Geom. 18 (2000), 447-475. 6 K. Watanabe; Plane domains which are spectrally determined II, Jour. Inequal. Appl. 7 (2002), 25-47. 7 S. Zelditch; Spectral determination of analytic bi-symmetric plane domains, Geom. Funct. Anal. 10 (2000), 628-677.
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