I need to prove the following proposition:

Any continuous linear functional on $H^1(\Omega)$ is of the form

$v\mapsto\displaystyle\int_\Omega\left\{\sum_{i=1}^nq_i\,\dfrac{\partial v}{\partial x_i}+q_0v\right\}$

with $q_i\in L^2(\Omega)$, $i=0,\dots n$.

Here $\Omega$ is an open bounded subset of $\mathbb{R}^n$ with a Lipschitz continuous boundary, and $H^1(\Omega)$ is the usual Sobolev space.

The book says that is a consequence from Banach Theorem, but I can't see the proof idea.

  • 1
    $\begingroup$ Try to see $H^1(\Omega)$ as a subset of $(L^2(\Omega))^{N+1}$ $\endgroup$ – Tomás Jan 4 '15 at 17:54
  • $\begingroup$ Sorry, I don't understand the hint. Can you help me with more details? $\endgroup$ – yemino Jan 4 '15 at 18:09
  • $\begingroup$ $H^1(\Omega)=\{v\in L^2(\Omega):\, \nabla u\in [L^2(\Omega)]^n\}$. I don't understant the hint because $H^1(\Omega)$ is an scalar space (it is not vector). $\endgroup$ – yemino Jan 4 '15 at 18:25
  • $\begingroup$ I tried to look the Hanh Banach theorem, but this theorem extends functional defined on a subspace to the whole space, so ? can't associate the Theorem with my problem $\endgroup$ – yemino Jan 4 '15 at 18:27
  • $\begingroup$ To give a function $u\in H^1(\Omega)$, is to give $N+1$ functions in $L^2(\Omega)$. $\endgroup$ – Tomás Jan 4 '15 at 18:37

Ok, here it goes. We assume $F$ is an linear bounded operator over $H^1(\Omega)$

Let $E$ denote the space of $N+1$ fold $L^2(\Omega)$, i.e., $E(\Omega):=(L^2(\Omega))^{N+1}$. Then the operator $T$, from $H^1(\Omega)\to E(\Omega)$ is defined by $T[u]=(u,\partial_1 u,\partial_2u,\ldots,\partial_Nu)$ and we have $T[u]\in E(\Omega)$. Take $G:=T(H^1(\Omega))$ and $S:=T^{-1}$. Then, the linear operator $L$ over $G$ defined as $L(h):=\left<F,S(h)\right>$ is continuous because $F$ is continuous.

Now is where we use Hanh-Banach extension theorem. We have $G$ is a subspace over $E$ and hence we could extend $L$ from $G$ to whole $E$. Hence, $L$ is now a linear continuous operator over $E$. To conclude, we recall Riesz representation for $L^p$ and we obtain $v_0$, $v_1$, $v_2$...$v_n\in L^2(\Omega)$ such that $$ L(h)=\int_\Omega h_1v_0+\int_\Omega h_2v_1+\cdots+\int_\Omega h_{n+1}v_N\,dx $$ In particular, if $h\in G$ we have $$ L(h)=\left<F,S(h)\right>=\sum_{i=1}^N\int_\Omega \partial_i uv_i\,dx+\int_\Omega uv_0\,dx $$ as you expected.

Remark I don't think the boundary of $\Omega$ will matter, i.e., any open set would be good. Also, if you have $H_0^1$ you could set $v_0=0$.

Remark2 You can just extend $L$ by means of continuity but not H-B-E. And also notice that the sequence $(v_i)$ may not unique.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.