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A heat trace anomaly on polygons

Published online by Cambridge University Press:  19 June 2015

RAFE MAZZEO  and
JULIE ROWLETT
RAFE MAZZEO
Affiliation:
Department of Mathematics, Stanford University, Building 380, Stanford, CA 94305, U.S.A. e-mail: [email protected]
JULIE ROWLETT
Affiliation:
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Göteborg, Sweden e-mail: [email protected]
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Abstract

Let Ω0 be a polygon in $\mathbb{R}$2, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that Ωε is a family of surfaces with ${\mathcal C}$∞ boundary which converges to Ω0 smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov [6], Kac [8] and McKean–Singer [13] recognised that certain heat trace coefficients, in particular the coefficient of t0, are not continuous as ε ↘ 0. We describe this anomaly using renormalized heat invariants of an auxiliary smooth domain Z which models the corner formation. The result applies to both Dirichlet and Neumann boundary conditions. We also include a discussion of what one might expect in higher dimensions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 2 , September 2015 , pp. 303 - 319
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

REFERENCES

[1] van dan Berg, M. and Srisatkunarajah, S. Heat equation for a region in $\mathbb{R}$ 2 with a polygonal boundary. J. London Math. Soc. 2 no. 37 (1988), 119–127.Google Scholar
[2] Cheeger, J. Spectral geometry of singular Riemannian spaces. J. Differential Geom. 18 (1984), 575–657.Google Scholar
[3] Dauge, M., Tordeux, S. and Vial, G. Self-similar perturbations near a corner: matching versus multiscale expansions for a model problem. Around the research of Vladimir Maz'ya. II, Int. Math. Ser. (N. Y.) 12 (Springer, New York, 2010), 95–134.Google Scholar
[4] Davies, E.B. Heat kernels and spectral theory. Cambridge Tracts in Math. 92 (Cambridge University Press, 1989).Google Scholar
[5] Ding, Y. Heat kernels and Green's functions on limit spaces, Comm. Anal. Geom., 10 no. 3 (2002), 475–514.Google Scholar
[6] Fedosov, B. Asymptotic formulas for eigenvalues of the Laplacian in a polyhedron. Doklady Akad. Nauk SSSR 157 (1964), 536–538.Google Scholar
[7] Guillarmou, C. and Hassell, A. Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conical manifolds, I. Math. Ann. 341 no. 4 (2008), 859–896, II. Ann. Inst. Fourier (Grenoble) 59 no. 4 (2009), 1553–1610.Google Scholar
[8] Kac, M. Can one hear the shape of a drum? Amer. Math. Monthly 73 (1966), 1–23.Google Scholar
[9] Kokotov, A. Polyhedral surfaces and the determinant of the Laplacian. Proc. Amer. Math. Soc. 141 no. 2 (2013), 725–735.Google Scholar
[10] Mazzeo, R. Elliptic theory of differential edge operators I. Comm. Partial Differential Equations 16 no. 10 (1991), 1615–1664.Google Scholar
[11] Mazzeo, R. Resolution blowups, spectral convergence and quasi-asymptotically conic spaces. Journées équations aux dérivées partielles Art. No. 8, (2006), 16 p.Google Scholar
[12] Melrose, R. The Atiyah–Patodi–Singer index theorem. Res. Not. Math. 4 (A K Peters, Ltd., 1993).Google Scholar
[13] McKean, H. P. Jr., and Singer, I. M. Curvature and the eigenvalues of the Laplacians. J. Differential Geom. 1 (1967), 43–69.Google Scholar
[14] Rowlett, J. Spectral geometry and asymptotically conic convergence. PhD. thesis (Stanford University, June 2006).Google Scholar
[15] Rowlett, J. Spectral geometry and asymptotically conic convergence. Comm. Anal. Geom. 16 no. 4 (2008), 735–798.Google Scholar
[16] Sher, D. The heat kernel on an asymptotically conic manifold. Anal. PDE 6 no. 7 (2013), 1755–1791.Google Scholar