%0 Journal Article
%J Communications in Mathematical Physics, Volume 315, Issue 1, September 2012, Pages 1-37
%D 2012
%T Non-uniqueness results for critical metrics of regularized determinants in four dimensions
%A Matthew Gursky
%A Andrea Malchiodi
%X The regularized determinant of the Paneitz operator arises in quantum gravity (see Connes 1994, IV.4.$\gamma$). An explicit formula for the relative determinant of two conformally related metrics was computed by Branson in Branson (1996). A similar formula holds for Cheeger's half-torsion, which plays a role in self-dual field theory (see Juhl, 2009), and is defined in terms of regularized determinants of the Hodge laplacian on $p$-forms ($p < n/2$). In this article we show that the corresponding actions are unbounded (above and below) on any conformal four-manifold. We also show that the conformal class of the round sphere admits a second solution which is not given by the pull-back of the round metric by a conformal map, thus violating uniqueness up to gauge equivalence. These results differ from the properties of the determinant of the conformal Laplacian established in Chang and Yang (1995), Branson, Chang, and Yang (1992), and Gursky (1997). We also study entire solutions of the Euler-Lagrange equation of $\log \det P$ and the half-torsion $\tau_h$ on $\mathbb{R}^4 \setminus {0}$, and show the existence of two families of periodic solutions. One of these families includes Delaunay-type solutions.
%B Communications in Mathematical Physics, Volume 315, Issue 1, September 2012, Pages 1-37
%I Springer
%G en
%U http://hdl.handle.net/1963/6559
%1 6488
%2 Mathematics
%4 1
%# MAT/05 ANALISI MATEMATICA
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%R 10.1007/s00220-012-1535-7
%0 Journal Article
%J Geometric and Functional Analysis 21 (2011) 1196-1217
%D 2011
%T New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces
%A Andrea Malchiodi
%A David Ruiz
%X We consider a singular Liouville equation on a compact surface, arising from the study of Chern-Simons vortices in a self dual regime. Using new improved versions of the Moser-Trudinger inequalities (whose main feature is to be scaling invariant) and a variational scheme, we prove new existence results.
%B Geometric and Functional Analysis 21 (2011) 1196-1217
%I Springer
%G en_US
%U http://hdl.handle.net/1963/4099
%1 305
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2010-11-10T10:27:54Z\\r\\nNo. of bitstreams: 1\\r\\nMalchiodi-Ruiz-74M.pdf: 192985 bytes, checksum: 61acb10ab3cde055824228920d16987a (MD5)
%R 10.1007/s00039-011-0134-7
%0 Journal Article
%J J. Funct. Anal. 180 (2001) 210-241
%D 2001
%T Non-compactness and multiplicity results for the Yamabe problem on Sn
%A Massimiliano Berti
%A Andrea Malchiodi
%B J. Funct. Anal. 180 (2001) 210-241
%I Elsevier
%G en
%U http://hdl.handle.net/1963/1345
%1 3110
%2 Mathematics
%3 Functional Analysis and Applications
%$ Made available in DSpace on 2004-09-01T12:56:35Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1999
%R 10.1006/jfan.2000.3699
%0 Journal Article
%J Ricerche Mat. 49 (2000), suppl., 169-176
%D 2000
%T A note on the scalar curvature problem in the presence of symmetries
%A Antonio Ambrosetti
%A Li YanYan
%A Andrea Malchiodi
%B Ricerche Mat. 49 (2000), suppl., 169-176
%I SISSA Library
%G en
%U http://hdl.handle.net/1963/1365
%1 3090
%2 Mathematics
%3 Functional Analysis and Applications
%$ Made available in DSpace on 2004-09-01T12:56:52Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1999