2009/01/31 by Rafe Mazzeo, Julie Rowlett · 1 citation
Mathematics · #math.DG #math.SP #msc:58J50
paper · pdf · doi:10.1017/s0305004115000365
published as Math. Proc. Camb. Phil. Soc. 159 (2015) 303-319 · Revision includes treatment of the Neumann problem and a discussion of the higher dimensional case; some new references
arxiv created 2009/08/20 · arxiv updated 2019/07/22
Let Ω0 be a polygon in \RR2, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that Ω_\e is a family of surfaces with \calC^∞ 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 \citeFe, Kac \citeK and McKean-Singer \citeMS recognized that certain heat trace coefficients, in particular the coefficient of t0, are not continuous as \e \searrow 0. We describe this anomaly using renormalized heat invariants of an auxiliary smooth domain Z which models the corner formation. The result applies both for Dirichlet and Neumann conditions. We also include a discussion of what one might expect in higher dimensions.