An improved Moser-Trudinger inequality involving the first non-zero Neumann eigenvalue with mean value zero in ${\mathbf{R}}^2$

Abstract

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 \[ \sup_{u} \int_Ω e^{2πu^2} dx < +\infty \] for arbitrary functions $u$ in $H^1(\Omega )$ satisfying ${\int }_{\Omega }udx=0$ and $\Vert \nabla u{\Vert }_2^2-\alpha \Vert u{\Vert }_2^2⩽1$. Furthermore, this supremum is attained by some function $u^{\ast }\in H^1(\Omega )$. 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).

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References in corpus (1)

• A sharp Adams inequality in dimension four and its extremal functions