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I came across the following question when I read chapter 17 of Hormander's book "Tha Analysis of Linear Partial Differential Operators", and the theorem is

Let $a_{jk}(x)$ be Lipschitz continuous in an open set $X\subset\mathbb{R}^n$, $a_{ij}=a_{ji}$, and assume that $(\Re a_{ij}(x))$ is positive definite. Then $$ \sum_{ij} D_j(a_{jk}D_ku)=f $$ has a solution $u\in H_{(2)}^{loc}(X)$ for every $f\in L_{loc}^2(X)$

The auther then says if we can show that $$ |(f,\phi)|\leq \|M \cdot\sum_{ij} D_j(\bar{a_{jk}}D_k\phi) \|_{L^2}, \quad \phi\in C_c^{\infty}(X) $$ for some positive continuous function $M$, then by Hahn-Banach theorem there exists some $g\in L^2$ $$ (f,\phi)=\left(g,M\cdot\sum_{ij} D_j(\bar{a_{jk}}D_k\phi)\right) $$ which inplies that the weak solution is $u=Mg$. what confuses me is how the Hahn-Banach theorem is used here to show the existence of $g$.

Thanks for your help

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This $\Sigma$ is the sum $\sum_{ij}$ \sum_{ij}? – Ilya Jun 5 '13 at 16:02
@Ilya,yes, thanks – sun Jun 5 '13 at 16:06
There is missing something in your equations. In the inequality and in the equality, you have to add the terms containing $\phi$. – Tomás Jun 5 '13 at 23:20
@Tomás, ah,sorry about that... – sun Jun 5 '13 at 23:45
Your notaation is different form that on the book. There he consider the product $D_j\overline{a}_{jk}D_k\phi$, instead of $D_j(\overline{a}_{jk} D_k\phi)$ – Tomás Jun 6 '13 at 0:00

2 Answers 2

I didn't solve it, but what I have may be helpful and it's too long for a comment.

Recall that $C_c^\infty$ is a dense linear subset of $L^2$. So by the Hahn-Banach theorem there is a norm preserving extension of $(f,\phi)$ to $(L^2)^*$. Then since $(L^2)^*$ is a Hilbert space the Riesz representation theorem tells us this extension has the form $\phi^{**}\longmapsto (g,\phi^{**})$ for some $g \in (L^2)^{**}$. Identifying $(L^2)^{**}=L^2$ then tells us $ (f,\phi) = (g,\phi) $ for all $\phi \in L^2$.

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Define the functional:

$$k(M Lw)=\int_{X} fw$$

where $L$ is the differential opeartor.

This is a bounded functional thanks to estimate you are assuming, also notice that you use the estimate to check that $k$ is well defined .

Then thanks to the Hahn-Banach theorem you can extendthe domain of this functional to the whole $L^{2}$ as nullUser mention.

Finally you use Riesz-Representation theorem to obtain the solution.

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