Abstract
AbstractWe prove a lower bound on the sharp Poincaré–Sobolev embedding constants for general open sets, in terms of their inradius. We consider the following two situations: planar sets with given topology; open sets in any dimension, under the restriction that points are not removable sets. In the first case, we get an estimate which optimally depends on the topology of the sets, thus generalizing a result by Croke, Osserman and Taylor, originally devised for the first eigenvalue of the Dirichlet–Laplacian. We also consider some limit situations, like the sharp Moser–Trudinger constant and the Cheeger constant. As a byproduct of our discussion, we also obtain a Buser-type inequality for open subsets of the plane, with given topology. An interesting problem on the sharp constant for this inequality is presented.
Funder
Università degli Studi di Ferrara
Publisher
Springer Science and Business Media LLC
Reference64 articles.
1. Adams, R.A.: Compact Sobolev imbeddings for pepper sets. J. Math. Anal. Appl. 27, 405–408 (1969)
2. Almgren, F.J., Lieb, E.H.: Symmetric decreasing rearrangement is sometimes continuous. J. Am. Math. Soc. 2, 683–773 (1989)
3. Bañuelos, R., Carroll, T.: Brownian motion and the fundamental frequency of a drum. Duke Math. J. 75, 575–602 (1994)
4. Bates, S.M.: Toward a precise smoothness hypothesis in Sard’s theorem. Proc. Am. Math. Soc. 117, 279–283 (1993)
5. Battaglia, L., Mancini, G.: Remarks on the Moser-Trudinger inequality. Adv. Nonlinear Anal. 2, 389–425 (2013)