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After introducing the notion of capacity in a general Hilbert space setting we look at the spectral bound of an arbitrary self-adjoint and semi-bounded operator . If is subjected to a domain perturbation the spectrum is shifted to the right. We show that the magnitude of this shift can be estimated in terms of the capacity. We improve the upper bound on the shift which was given in Capacity in abstract Hilbert spaces and applications to higher order differential operators (Comm. P. D. E., 24:759–775, 1999) and obtain a lower bound which leads to a generalization of Thirring’s inequality if the underlying Hilbert space is an -space. Moreover, a similar capacitary upper bound for the second eigenvalue is established. The results are finally applied to higher-order partial differential operators.
@incollection{JEDP_1999____A8_0,
author = {Noll, Andr\'e},
title = {Domain perturbations, capacity and shift of eigenvalues},
booktitle = {},
series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {8},
pages = {1--10},
publisher = {Universit\'e de Nantes},
year = {1999},
doi = {10.5802/jedp.552},
mrnumber = {2000h:47023},
zbl = {01810581},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.5802/jedp.552/}
}
TY - JOUR AU - Noll, André TI - Domain perturbations, capacity and shift of eigenvalues JO - Journées équations aux dérivées partielles PY - 1999 SP - 1 EP - 10 PB - Université de Nantes UR - http://geodesic.mathdoc.fr/articles/10.5802/jedp.552/ DO - 10.5802/jedp.552 LA - en ID - JEDP_1999____A8_0 ER -
Noll, André. Domain perturbations, capacity and shift of eigenvalues. Journées équations aux dérivées partielles (1999), article no. 8, 10 p.. doi: 10.5802/jedp.552
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