Hi,

I need an inequality similar to this one for bounded domain [0,L]. http://img94.imageshack.us/img94/3166/screenshot1qy.png

My u(x) is not 0 on the boundary.

I will appreciate if you can help me about this question,

Edit: More precisely, is the following the statement true?

For $F\in C^1[0,L]$ with $F(0)=0$ or $\int_0^L F(x)dx=0$, one has the estimate $$ \int_0^L|F(x)|^px^{\beta}dx\leq \left(\frac{p}{\beta+1}\right)^p\int_0^L|F'(x)|^px^{\beta+p}dx $$