Fractional conformal Laplacians and fractional Yamabe problems

By María del Mar González, J. Qing. 2010

157 citations recorded by the source corpus.

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 Gonzalez, 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. © 2013 Mathematical Sciences Publishers.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/80214d79c7e5836ee90152c7d0e581808aa36f11

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Citations in the corpus

On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions (2011, background): “Fractional Yamabe problems about finding constant σ -curvatures on the conformal infinity of given asymptotically hyperbolic manifolds have been studied by González, Mazzeo and Sire [54], González and Qing [55] and Qing and Raske [87].”
An extension problem for the CR fractional Laplacian (2013, result): “γ m−γ , on M. From the variational point of view, we are looking for minimizers of the functional Iγ[f] = ´ M fP θ γfθ∧dθn ´ M |f|2 ∗θ∧dθn 2 2 , for 2∗ = 2m m−γ. Motivated by the Riemannian case from [32], one may ﬁnd instead minimizers of the extension functional Iγ[u] = ´ X q m−1|∇u|2 g+ dvolg+ −s”
A fractional Yamabe flow and some applications (2011, background): “een several work on these conformally invariant equations of fractional order and prescribing σ-curvatures problem (fractional Yamabe problem and fractional Nirenberg problem), see, e.g., [68], [39], [40], [51] and [52]. In this paper we study some ﬂow of this fractional order curvature Rg σ associa”
Existence theorems of the fractional Yamabe problem (2016, methodology): “This problem is referred to be thefractional Yamabe problemor theγ-Yamabe problem, and it was introduced and investigated by González-Qing [19] and González-Wang [20].”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, background): “It is possible then to formulate fractional Yamabe-type problems for the conformally covariant operators Pγ , [138].”
Some constructions for the fractional Laplacian on noncompact manifolds (2012, background): “It is possible then to formulate fractional Yamabe-type problems for Pγ , as considered in [31], where the main ingredients needed in the proof are a Hopf’s maximum principle, elliptic estimates and a sharp Sobolev trace inequality.”
On Fractional GJMS Operators (2014, methodology): “In order to motivate our approach, let us first rederive using our language of smooth metric measure spaces a result of González and Qing [12] on sufficient conditions for the positivity of the first eigenvalue of P2γ .”
Delaunay-type singular solutions for the fractional Yamabe problem (2015, background): “For instance, as a consequence of our construction one obtains the non-uniqueness of the solutions for the fractional Yamabe equation (see [17] for an introduction to this problem) in the positive curvature case, since it gives different conformal metrics on S(1)(L) × Sn−1 that have constant fractio”
ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH THE FRACTIONAL LAPLACIAN (2013, background): “We remark that a similar argument to our proof appeared in [23].”