A sharp Trudinger–Moser inequality on any bounded and convex planar domain

By Guozhen Lu, Qiaohua Yang. 2015

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Wang and Ye conjectured in (Adv Math 230:294–320, 2012): Let$$\Omega $$Ωbe a regular, bounded and convex domain in$$\mathbb {R}^{2}$$R2. There exists a finite constant$$C({\Omega })>0$$C(Ω)>0such that$$\begin{aligned} \int _{\Omega }e^{\frac{4\pi u^{2}}{H_{d}(u)}}dxdy\le C(\Omega ),\quad \forall u\in C^{\infty }_{0}(\Omega ), \end{aligned}$$∫Ωe4πu2Hd(u)dxdy≤C(Ω),∀u∈C0∞(Ω),where$$H_{d}=\int _{\Omega }|\nabla u|^{2}dxdy-\frac{1}{4}\int _{\Omega }\frac{u^{2}}{d(z,\partial \Omega )^{2}}dxdy$$Hd=∫Ω|∇u|2dxdy-14∫Ωu2d(z,∂Ω)2dxdyand$$d(z,\partial \Omega )=\min \limits _{z_{1}\in \partial \Omega }|z-z_{1}|$$d(z,∂Ω)=minz1∈∂Ω|z-z1|. The main purpose of this paper is to confirm that this conjecture indeed holds for any bounded and convex domain in $$\mathbb {R}^{2}$$R2 via the Riemann mapping theorem (the smoothness of the boundary of the domain is thus irrelevant). We also give a rearrangement-free argument for the following Trudinger–Moser inequality on the hyperbolic space $$\mathbb {B}=\{z=x+iy:|z|=\sqrt{x^{2}+y^{2}}<1\}$$B={z=x+iy:|z|=x2+y2<1}: $$\begin{aligned} \sup _{\Vert u\Vert _{\mathcal {H}}\le 1} \int _{\mathbb {B}}(e^{4\pi u^{2}}-1-4\pi u^{2})dV=4\sup _{\Vert u\Vert _{\mathcal {H}}\le 1}\int _{\mathbb {B}}\frac{(e^{4\pi u^{2}}-1-4\pi u^{2})}{(1-|z|^{2})^{2}}dxdy< \infty , \end{aligned}$$sup‖u‖H≤1∫B(e4πu2-1-4πu2)dV=4sup‖u‖H≤1∫B(e4πu2-1-4πu2)(1-|z|2)2dxdy<∞,by using the method employed earlier by Lam and the first author (Adv Math 231(6):3259–3287, 2012, J Differ Equ 255:298–325, 2013), where $$\mathcal {H}$$H denotes the closure of $$C^{\infty }_{0}(\mathbb {B})$$C0∞(B) with respect to the norm $$\begin{aligned} \Vert u\Vert _{\mathcal {H}}=\int _{\mathbb {B}}|\nabla u|^{2}dxdy-\int _{\mathbb {B}}\frac{u^{2}}{(1-|z|^{2})^{2}}dxdy. \end{aligned}$$‖u‖H=∫B|∇u|2dxdy-∫Bu2(1-|z|2)2dxdy.Using this strengthened Trudinger–Moser inequality, we also give a simpler proof of the Hardy–Moser–Trudinger inequality obtained by Wang and Ye (Adv Math 230:294–320, 2012).

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/40cf5b1d9b345d7e580907ac1d3c6b86159da3a5

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In Applied Mathematics

  1. Guozhen Lu
  2. Qiaohua Yang

In Computational Theory and Mathematics

  1. Guozhen Lu
  2. Qiaohua Yang

In Geometry and Topology

  1. Qiaohua Yang
  2. Guozhen Lu

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