On the local solvability of the Nirenberg problem on \mathbb S^2

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June 2010, 28(2): 607-615. doi: 10.3934/dcds.2010.28.607

On the local solvability of the Nirenberg problem on $\mathbb S^2$

1.	Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, United States
2.	Department of Mathematics, Rutgers University, Hill Center, 110 Frelinghuysen Rd., Piscataway, NJ 08854

Received January 2010 Revised March 2010 Published April 2010

Abstract
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We present some results on the local solvability of the Nirenberg problem on $\mathbb S^2$. More precisely, an $L^2(\mathbb S^2)$ function near $1$ is the Gauss curvature of an $H^2(\mathbb S^2)$ metric on the round sphere $\mathbb S^2$, pointwise conformal to the standard round metric on $\mathbb S^2$, provided its $L^2(\mathbb S^2)$ projection into the the space of spherical harmonics of degree $2$ satisfy a matrix invertibility condition, and the ratio of the $L^2(\mathbb S^2)$ norms of its $L^2(\mathbb S^2)$ projections into the the space of spherical harmonics of degree $1$ vs the space of spherical harmonics of degrees other than $1$ is sufficiently small.

Keywords: Nirenberg problem, prescribing Gaussian curvature, local solvability..

Mathematics Subject Classification: Primary: 35J60; 58J05; 53A3.

Citation: Zheng-Chao Han, YanYan Li. On the local solvability of the Nirenberg problem on $\mathbb S^2$. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 607-615. doi: 10.3934/dcds.2010.28.607

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