Vol. 6, No. 7, 2013

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Fractional conformal Laplacians and fractional Yamabe problems

María del Mar González and Jie Qing

Vol. 6 (2013), No. 7, 1535–1576

Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate elliptic boundary value problems observed by Chang and González, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopf-type maximum principle together with interplay between analysis of weighted trace Sobolev inequalities and conformal structure of the underlying manifolds, which extends the phenomena displayed in the classic Yamabe problem and boundary Yamabe problem.

fractional Laplacian, conformal geometry, Yamabe problem
Mathematical Subject Classification 2010
Primary: 35J70, 53A30, 35R11
Received: 18 September 2011
Revised: 5 September 2012
Accepted: 18 October 2012
Published: 27 December 2013
María del Mar González
ETSEIB, Departament de Matemàtica Aplicada I
Universitat Politècnica de Catalunya
Av. Diagonal, 647
08028 Barcelona
Jie Qing
Department of Mathematics
University of California, Santa Cruz
4111 McHenry
Santa Cruz, CA 95064
United States