Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An improved Moser-Trudinger inequality involving the first non-zero
  Neumann eigenvalue with mean value zero in $\mathbf R^2$   Let $\Omega$ be a smooth bounded domain in $\mathbf R^2$ and
$\lambda^{\mathsf N} (\Omega)$ the first non-zero Neumann eigenvalue of the
operator $-\Delta$ on $\Omega$. In this paper, for any $\gamma \in [0,
\lambda^{\mathsf N} (\Omega) )$, we establish the following improved
Moser-Trudinger inequality \[ \sup_{u} \int_{\Omega} e^{2\pi u^2} dx < +\infty
\] for arbitrary functions $u$ in $H^1(\Omega)$ satisfying $\int_\Omega u dx
=0$ and $\|\nabla u\|_2^2 -\alpha \|u\|_2^2 \leqslant 1$. Furthermore, this
supremum is attained by some function $u^*\in H^1(\Omega)$. This strengthens
the results of Chang and Yang (J. Differential Geom. 27 (1988) 259-296) and of
Lu and Yang (Nonlinear Anal. 70 (2009) 2992-3001).

11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
20
The categories of this paper are: acc-phys
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph