Htykuut

I have to give an example (just construct not very formally really) of a function $f in H^1(Omega)$ that:

1. $Omega = (0,1)times(0,1), Gamma_1={(x,0);0leq xleq 1}cup{(y,0);0leq yleq 1}, Gamma_2=partialOmegabackslash Gamma_1$

2. $fin C^0(overline{Omega}), fnotin C^1(overline{Omega})$, $f in H^1(Omega)$ contains the continous function but does not contain the continously differentiable one.

3. $f neq 0$, the function does not totally equal to zero

4. $tr(f) neq 0$ on $Gamma_1$ and $tr(f)=0$ on $Gamma_2$ - while trace is not totally equal to zero on $Gamma_1$, it is trivial on the remaining part of the boundary.

I also have to compute the weak (distributional) derivative of f.

The problem is, I have no idea how to give an example of such a function and therefore construct one. I do know how to compute distributional derivatives, if I had such a function I could probably prove it fulfills these requirements, but constructing one is beyond my ability.

I get that $H^1(Omega)$ is a Hilbert space that is differentiable (in sense of having weak derivative) on subset $Omega in mathbb R^n$ and derivative of function is not continous on $Omega$, and that trace can be shown to be different than 0 from right computation. If I understand correctly, $Gamma_1$ and $Gamma_2$ are parts of $partial Omega$, which is the boundary of $Omega$. Neverthess as I stated before I don't know how to construct the function and show those things. Can you help me?