There exist a function $u(x)\in C([0,1]),~ u(0)=u(1)=0$ such that $u \notin H_{0}^{1}((0,1))$?

Question

There exist a function $u(x)\in C([0,1]),~ u(0)=u(1)=0$ such that $u \notin H_{0}^{1}((0,1))$? Justify your answer!

Thanks.

Have you tried to find a differentiable function with derivative in $(0,1)$ not in $L^2([0,1])$? — AppliedSide, Apr 29 '13 at 19:42

score 2 · Accepted Answer · answered Apr 29 '13 at 19:43

2

Try checking if all of the properties of $H^1_0((0,1))$ are met for the following function: $$ f(x) = x\sin\frac{\pi}{x} $$

Intuition: construct an example of a continuous function, it oscillates so much that its derivative may not be $L^2$-integrable.

answered Apr 29 '13 at 19:43

Shuhao Cao

3

Another example would be $f(x)=\sqrt{x}-x$, where the derivative blows up too much. – AppliedSide Apr 29 '13 at 19:52
@AppliedSide That should be an answer. – Shuhao Cao Apr 29 '13 at 19:58

1 Answers1