SMS scnews item created by Daniel Daners at Wed 24 Jul 2013 1425
Type: Seminar
Distribution: World
Expiry: 29 Jul 2013 Calendar1: 29 Jul 2013 1400-1500 CalLoc1: AGR Carslaw 829
Auth: daners@como.maths.usyd.edu.au
PDE Seminar
Non-Positivity of the semigroup generated by the Dirichlet-to-Neumann operator
Daners
Daniel Daners University of Sydney Mon 29 July 2013 2-3pm, Carslaw 829 (AGR)
Abstract
Let be a bounded open set with smooth
boundary, and let . The Dirichlet-to-Neumann
operator is a closed operator on
defined as follows. Given solve the
Dirichlet problem
A solution exists if is not an eigenvalue of
with Dirichlet boundary conditions. If is smooth enough we define
where is the outer unit normal to . Let
be the strictly ordered
Dirichlet eigenvalues of on . It was shown by Arendt
and Mazzeo that is positive and irreducible if
. The question left open was whether or not the
semigroup is positive for any . The aim of this talk
is to explore this question by explicitly computing the semigroup for
the disc in . The example demonstrates some new phenomena:
the semigroup can change from not positive to positive
between two eigenvalues. This happens for
. Moreover, it is possible that
is positive for large , but not for small . The
occurrence of such eventually positive semigroups seems to be new. See preprint.
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