Dual space of Sobolev spaces induced by Elliptic equations

Question

Given a domain $\Omega \subset \mathbb{R}^n$, it is well-known that the dual space of the Sobolev space $W_0^{1,2}(\Omega)$ (also denoted by $H^1_0(\Omega)$ by many people) is, by definition, the space $W^{-1,2}(\Omega)$. Also, since $W_0^{1,2}(\Omega)$ is a Hilbert space; Therefore its dual is isomorphic to itself, by the Riesz representation theorem. Thus there is an isomorphism, say $\mathcal {E}$, between $W_0^{1,2}(\Omega)$ and $W^{-1,2}(\Omega)$.

One such isomorphism that interest to me is the one induced by an elliptic operator. Let $L= - \mathrm{div} (A \nabla)$ be an uniformly elliptic operator; that is, $\lambda |\xi|^2 \leq \langle A(X) \xi, \xi \rangle \leq \lambda^{-1} |\xi|^2$ for some $\lambda >0$, all $X \in \Omega$ and all non-zero $\xi \in \mathbb{R}^n$. Then $L$ would induce a duality pair $\langle u,v \rangle$ for $u \in W_0^{1,2}(\Omega)$ and $v \in W^{-1,2}(\Omega)$, and hence an isomorphism, say $\mathcal{E}_L$, between the two mentioned spaces.

I want to know what is the true definition of this duality pair, and the resulting isomorphism. It seems that, this maybe in the connected to some elliptic PDE's. Given $v \in W^{-1,2}(\overline{\Omega})$, there exists a unique solution $u \in W_0^{1,2}(\Omega)$ to the elliptic equation

$ \begin{cases} u+Lu=0,& \hskip 2mm \text {on} \hskip 2mm \Omega \\ u=v,& \hskip 2mm \text {on} \hskip 2mm \partial \Omega \end{cases}.$

But I'm not sure if this the right formulation of the mapping $v \mapsto u$; and, even this is true, what is the true definition of duality pairing $\langle u,v \rangle$ and the associated isomorphism $\mathcal{E}_L$ in this case.

Any reference would be appreciated. There is a related question here (especially example 3 therein).

The relation between $u$ and $v$ is $Lu=f$ in $\Omega$. Then $v(\phi) = \langle Lu,\phi\rangle$ for all $\phi\in H^1_0$ — daw, Jun 07 '23 at 11:52
@daw Thanks for comment. It is a bit confusing to me, though. What is $f$ in $Lu=f$? — XIII, Jun 07 '23 at 14:01

score 1 · Answer 1 · answered Jun 08 '23 at 16:47

This is an answer to the above question which I report (almost word by word) from section 6.2. of Evans's book (page 320).

By a modification of the argument for the proof of Theorem 3, we have the following:

$\textbf{Proposition.}$ There exists $\gamma >0$, such that for any $\mu \geq \gamma$ and every functions $f^i \in L^2 (\Omega)$, $i=0,1, \ldots , n$ (with $\mathbf{f}= (f_1 , \ldots, f_n)$), there exists a unique solution $u$ to the equation

$\begin{cases} Lu+\mu u=f^0- \sum_{i=1}^n f^i_{x_i}= f^0- \mathrm{div} (\mathbf{f}) ,& \hskip 2mm \text {in} \hskip 2mm \Omega \\ u=0,& \hskip 2mm \text {on} \hskip 2mm \partial \Omega \end{cases}.$

Indeed, it is enough to note $\langle f,v \rangle = \int_\Omega [ f^0 v+ \sum_{i=1}^n f^i v_{x_i}] dx= \int_\Omega [ f^0 v + \mathbf{f} \cdot \nabla v] dx$ is a bounded linear functional on $H^1_0(\Omega)$. In particular, we deduce that the mapping $L_\mu:= L+ \mu I: H^1_0(\Omega) \to H^{-1}(\Omega)$ is an isomorphism.

Evans, Lawrence C., Partial differential equations, Graduate Studies in Mathematics 19. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4974-3/hbk). xxi, 749 p. (2010). ZBL1194.35001.

Dual space of Sobolev spaces induced by Elliptic equations

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