Help me please with this question:
Is it true that for all $N>0$ exists $u \in C^{1}[0,1]$ such that $u(0)=0$ and $\frac{\int_{0}^{1}u^2(x)dx}{\int_{0}^{1}\left [ u'(x) \right ]^2dx}>N$; what happens without assuming that $u(0)=0$?
Thanks a lot!
Help me please with this question:
Is it true that for all $N>0$ exists $u \in C^{1}[0,1]$ such that $u(0)=0$ and $\frac{\int_{0}^{1}u^2(x)dx}{\int_{0}^{1}\left [ u'(x) \right ]^2dx}>N$; what happens without assuming that $u(0)=0$?
Thanks a lot!
If $u\in C^1[0,1]$ satisfies $u(0)=0$ then $\int_0^1u(x)^2dx=\int_0^1\left(\int_0^xu'(t)dt\right)^2dx\leq \int_0^1x\int_0^x(u'(t))^2dtdx\leq \frac 12\int_0^1(u'(t))^2dt,$ hence for $K>1/2$ the inequality you want doesn't hold.
If we don't require $u(0)=0$, then take functions like $u(x)=x+C$ to get want we want.