Let Ω be a smooth bounded open subset of Rn. Does there exist A=A(Ω) with the property that for any f∈C∞(ˉΩ) with f=0 on ∂Ω, ∫Ω|f(x)|dx≤A∫Ω|∇f(x)|dx? I think that this is known as some version of “Poincare’s inequality”.
I found the answer at http://anhngq.wordpress.com/2010/02/22/the-poincare-inequality/. By Holder’s Inequality and Poincare’s Inequality for W1,p0 spaces, there exist A(Ω),B(Ω)>0 such that