On the Best Constant in the Moser-Onofri-Aubin Inequality

Abstract

Let $S^2$ be the 2-dimensional unit sphere and let $J_{\alpha }$ denote the nonlinear functional on the Sobolev space $H^{1,2}(S^2)$ defined by $$J_{\alpha }(u)=\frac{\alpha }{4}{\int }_{S^2}|\nabla u{|}^{2}d\omega +{\int }_{S^2}ud\omega -\ln {\int }_{S^2}{e}^{u}d\omega ,$$ where $d\omega$ denotes Lebesgue measure on $S^2$, normalized so that ${\int }_{S^2}d\omega =1$. Onofri had established that $J_{\alpha }$ is non-negative on $H^1(S^2)$ provided $\alpha \ge 1$. In this note, we show that if $J_{\alpha }$ is restricted to those $u\in H^1(S^2)$ that satisfy the Aubin condition: \int_{S^2}e^u x_j dw=0\quad\text{for all}1\leq j\leq 3, then the same inequality continues to hold (i.e., $J_{\alpha }(u)\ge 0$) whenever $\alpha \ge 2/3-{\epsilon }_0$ for some ${\epsilon }_0>0$. The question of Chang-Yang on whether this remains true for all $\alpha \ge 1/2$ remains open.

arXiv comment8 pages. Updated versions - if any - can be downloaded at http://www.birs.ca/~nassif/

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