Representation of Dirac function over Sobolve space

Question

Let $1<p<\infty$ and $q=\frac{p}{p-1}$. We know that $l\in (W^{k,p}(\Omega))'$ ($l$ is a continuous functional over $W^{k,p}$(\Omega)) if and only if there exits $\{f_\alpha\}\subset L^q(\Omega)$ such that $$ l(v)=\sum_{|\alpha|\leq k}\int_{\Omega} f_\alpha(x) \partial^\alpha v(x) dx\quad\forall\, v\in W^{k,p}(\Omega). $$ Now let us consider the simplest case, say $\Omega=(-1,1)$. Becase $H^{1}(-1,1)$ can be embedded into the space $C^{1/2}$, the dirac function $\delta\in (H^{1}(-1,1))'$. An natrual question arises? What is the $(f_0,f_1)\in L^2(-1,1)^2$ such that $$ v(0)=\int_{-1}^1 f_0(x)(x) v(x)+f_1(x) v'(x) dx \,\,\forall\, v\in H^1(-1,1) $$ If we set $H=\begin{cases} 1,\,\, x\geq 0,\\ 0,\,\, x<0 \end{cases}$, then we can find that $$ v(0)= -\int_0^1 H(x)v'(x) dx ,\,\forall\, v\in H^1_0(-1,1). $$ Update Someone has given the answer to the first dimension. However the general domain in higher-dimensional? How to find $f_\alpha$ such that $$ v(0)= -\int_\Omega H(x)v'(x) dx ,\,\forall\, v\in H^k(\Omega). $$ with $k>\frac{n}{2}$.

Answer 1

Let us take $f_1=\begin{cases} x+1,\,\, x<0\\ 0,\,\, x\geq 0\end{cases}\,\, f_0=\begin{cases} 1,\,\, x<0\\ 0,\,\, x\geq 0\end{cases}$. Then we can obtain by integration by parts that
$$ v(0)=\int_{-1}^1 f_0(x) v(x)+f_1(x) v'(x)dx\,\,\, \forall v\in H^1(-1,1) $$