Conditions au bord spectrales et formules de trace
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.

@article{SEDP_2001-2002____A15_0,
author = {Grubb, Gerd},
title = {Conditions au bord spectrales et formules de trace},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
note = {talk:15},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2001-2002},
language = {fr},
url = {http://archive.numdam.org/item/SEDP_2001-2002____A15_0/}
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Grubb, Gerd. Conditions au bord spectrales et formules de trace. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2001-2002), Exposé no. 15, 12 p. http://archive.numdam.org/item/SEDP_2001-2002____A15_0/

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