Mathematics – Analysis of PDEs
Scientific paper
2009-10-05
Mathematics
Analysis of PDEs
8 pages. Updated versions - if any - can be downloaded at http://www.birs.ca/~nassif/
Scientific paper
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 $$ J_\alpha(u) = \frac{\alpha}{4}\int_{S^2}|\nabla u|^2 d\omega + \int_{S^2} u d\omega -\ln \int_{S^2} e^{u} d\omega, $$ 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 $\alpha \geq 1$. 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., $J_\alpha (u)\geq0$) whenever $\alpha \geq {2/3}-\epsilon_0$ for some $\epsilon_0>0$. The question of Chang-Yang on whether this remains true for all $\alpha \geq {1/2}$ remains open.
Ghoussoub Nassif
Lin Chang-Shou
No associations
LandOfFree
On the Best Constant in the Moser-Onofri-Aubin Inequality does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the Best Constant in the Moser-Onofri-Aubin Inequality, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the Best Constant in the Moser-Onofri-Aubin Inequality will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-357094