Prof. Bernd Ammann und Mitarbeiter, Zimmer 119

Lashi Bandara, Univ. Potsdam

First-order elliptic boundary value problems beyond self-adjoint induced boundary operators

Monday, 3.6.2019, 14.15 in M104

The Bär-Ballmann framework is a comprehensive framework to consider elliptic boundary value problems (and also their index theory) for first-order elliptic operators on manifolds with compact and smooth boundary. A fundamental assumption in their work is that the induced operator on the boundary is symmetric. Many operators satisfy this requirement including the Hodge-Dirac operator as well as the Atiyah- Singer Dirac operator. Recently, there has been a desire to study more general operators with the quintessential example being the Rarita- Schwinger Dirac operator, which is an operator that fails to satisfy this hypothesis.

In this talk, I will present recent work with Bär where we dispense the symmetry assumption and consider general elliptic operators. The ellipticity of the operator still allows us to understand the spectral theory of the induced operator on the boundary, modulo a lower order additive perturbation, as bi-sectorial operator. We use a mixture of methods coming from pseudo-differential operator theory, bounded holomorphic functional calculus, semi-group theory as well as methods arising from the resolution of the Kato square root problem to recover many of the results of the Bär-Ballman framework.

If time permits, I will also touch on the non-compact boundary case, and potential extensions of this to the Lp-setting and Lipschitz boundary.

Bernd Ammann, 04.08.2019
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