Conditions au bord spectrales et formules de trace

Grubb, Gerd¹

¹ Copenhagen Univ. Math. Dept., Universitetsparken 5, DK-2100 Copenhague, Danemark.

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2001-2002), Exposé no. 15, 12 p.

Résumé

The lecture presents current results on heat trace expansions, and the related resolvent trace and zeta function expansions, for elliptic operators with boundary conditions on $n$-dimensional compact manifolds. As a background, we recall the set-up of elliptic differential operators with differential boundary conditions having heat trace expansions in powers ${\sum }_{k\ge -n}{c}_k{t}^{k/m}$. Then we consider the spectral boundary conditions of Atiyah, Patodi and Singer for Dirac-type first-order operators, leading to expansions with additional logarithmic terms ${\sum }_{k\ge 0}{c}_k^{\prime }{t}^{k/2}logt$ (joint work with Seeley 1995) ; an extension to “well-posed” problems is included in a general study of pseudo-normal boundary conditions (1999). New results are presented on the vanishing or stability of the $c_k^{\prime }$-coefficients ; special features appear when $n$ is odd. Finally, we study the pseudodifferential projection boundary conditions proposed by Vassilevich (2001) in string- and brane-theory, showing that they too have heat expansions with log-terms, under suitable hypotheses. In all cases, the lowest log-coefficient $c_0^{\prime }$ vanishes, which assures that the zeta function is regular at 0.