I am working with a subset of the space $H^1(0,1)$ and I would like to know whether the Poincare's Inequality would remain true for this space. I know that in higher dimensions the validity of Poincare's inequality in $H_0^1(0,1)$ can be extended to the case of traces vanishing only on a portion of the boundary as long as this portion has nonzero Hausdorff Measure. However, I have never found (or maybe I do not know where to look for it) a proof or counter--example for the case I am considering here.
I am not a specialist in inequalities, but what I had in mind was the following. If we consider a function $f \in V_0$ with $$V_0 = \{u \in H^1(0,1); u(1)=0\}$$ then given $x \in (0,1)$ we can write $u(1) - u(x) = \int_x^1 u'(x)dx$ and then taking absolute value both sides we get $$\|u\|_\infty^2 \leqslant \|u'\|_{L^1}^2 \leqslant \|u'\|_{L^2}^2$$ and finally we use that $L^\infty(0,1)$ is continuously embedded in $L^2(0,1).$
Are there any bubbles in this arguments? Usually I let some pretty simple mistakes get in the middle of my arguments, so I wanted to be sure it works.
Other than having one less boundary condition, are there any major property of $H_0^1(0,1)$ that does not work for $V_0$? Like, I want $V_0$ to be closed (I think it is) and dense in $L^2$ (which I also think it is because $H_0^1 \subset V_0 \subset H^1.$
If anybody knows a book or any material that deal with properties of spaces like that, I would really appreciate if they could share.
I appreciate the help in advance. :)