Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Best Constant in the Moser-Onofri-Aubin Inequality   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.

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The categories of this paper are: nucl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: supr-con
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The categories of this paper are: funct-an
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The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ex
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The categories of this paper are: chao-dyn
30
The categories of this paper are: q-alg
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The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph
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The categories of this paper are: econ
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The categories of this paper are: math-ph