Conditions au bord spectrales et formules de trace

Grubb, Gerd

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

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.

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