An improved singular Trudinger-Moser inequality in dimension two

Anfeng YUAN; Zhiyong HUANG

Makale detayları

An improved singular Trudinger-Moser inequality in dimension two

Yazarlar:

Anfeng YUAN | Zhiyong HUANG

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 2016 Vol.40 Issue 4, pp.874-883

Belge Tipi:

Makale

Yayın Tarihi:

7/2016

Belge Dili:

Türkçe

Document ID (Org No.):

2000746953

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098

Özet:

 

Let Ω⊂R2Ω⊂R2\Omega\subset\mathbb{R}^2 be a smooth bounded domain  and W1,20(Ω)W01,2(Ω)W_0^{1,2}(\Omega) be the usual Sobolev space.  Let ββ\beta, 0≤β10≤β10\leq\beta1,  λp,β(Ω)=infu∈W1,20(Ω),u≢0∥∇u∥22/∥u∥2p,β,λp,β(Ω)=infu∈W01,2(Ω),u≢0‖∇u‖22/‖u‖p,β2,\lambda_{p,\beta}(\Omega)=\inf_{u\in W_0^{1,2}(\Omega),\,u\not\equiv 0}{\|\nabla u\|_2^2}/{\|u\|_{p,\beta}^2},   where ∥⋅∥2‖⋅‖2\|\cdot\|_2  denotes the standard L2L2L^2-norm in ΩΩ\Omega and ∥u∥p,β=(∫Ω|x|−β|u|pdx)1/p‖u‖p,β=(∫Ω|x|−β|u|pdx)1/p\|u\|_{p,\beta}=({\int_{\Omega}|x|^{-\beta}|u|^pdx})^{1/p}.  Suppose that γγ\gamma satisfies \fγ4π+\fβ2=1\fγ4π+\fβ2=1\f{\gamma}{4\pi}+\f{\beta}{2}=1.  Using a rearrangement argument, the author proves that  supu∈W1,20(Ω),∥∇u∥2≤1∫Ω|x|−βeγu2≤(1+α∥u∥2p,β\ri)dxsupu∈W01,2(Ω),‖∇u‖2≤1∫Ω|x|−βeγu2≤(1+α‖u‖p,β2\ri)dx\sup_{u\in W_0^{1,2}(\Omega), \|\nabla u\|_2\leq 1}\int_{\Omega} |x|^{-\beta}e^{\gamma u^2 \le(1+\alpha\|u\|_{p,\beta}^2\ri)  }dx  is finite for any αα\alpha, $0\leq\alpha.

Makale detayları

An improved singular Trudinger-Moser inequality in dimension two

Yazarlar: Anfeng YUAN | Zhiyong HUANG

Yayıncı: TUBITAK

Dergi: Turkish Journal of Mathematics 2016 Vol.40 Issue 4, pp.874-883

Belge Tipi: Makale

Yayın Tarihi: 7/2016

Belge Dili: Türkçe

Document ID (Org No.): 2000746953

Sıklık: İki Aylık

eISSN: 1303-6149

pISSN: 1300-0098

Sayı

Yazarlar:

Anfeng YUAN | Zhiyong HUANG

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 2016 Vol.40 Issue 4, pp.874-883

Belge Tipi:

Makale

Yayın Tarihi:

7/2016

Belge Dili:

Türkçe

Document ID (Org No.):

2000746953

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098