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Context:

I am reading about Sobolev spaces and the Poisson equation from Eberhard Zeidler's Applied Functional Analysis book/article, and a key tool seems to be what Zeidler calls the "Main theorem on quadratic variational problems"(I believe this refers to the Lax Milgram theorem).

I've seen how the Poisson boundary value problem can be converted to a variational problem $a(u,v) = b(v)$ for all $v\in C^\infty_0(G\subset \mathbb{R}^n)$. In particular, $a = \int_G \sum_{j=1}^n \frac{\partial}{\partial x_j}u\frac{\partial}{\partial x_j}v dx$

I'm trying to understand this material better by getting a feel for what symmetric, coercive, continuous bilinear forms are like and how they ended up being related to partial differential equations.

Question:

In order to better understand this, I'm trying to understand two things

(1) What are the possible symmetric bilinear continuous functions $a:X \times X \rightarrow \mathbb{R}$. To be more concrete, for bounded linear functions $F$ on $L^2$, we know that $F(g) = \int_X fg dx $ for some function $f\in L^2(X)$. Is there a similar theorem characterizing the bilinear functions of $L^2$ or $W^2$.

(2) Is there a larger family of partial differential equations that can be converted into the form $a(u,v) = b(v)$ in the same way that the Poisson equation can be? Part of the reason I want to know this is just to have more reasonably motivated examples of bilinear forms $a$.

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    $\begingroup$ (1): You can identify the bounded bilinear form $a$ with the bounded linear operator $A : X \to X^*$, $x \mapsto (y \mapsto a(x,y))$. $\endgroup$ – gerw Feb 16 '16 at 14:37
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    $\begingroup$ One characterization for continuous (bounded) bilinear forms is: If $a(.,.): X\times X\to\mathbb R$, $a$ is bounded: $a(u,v)\leq C_2\|u\|\|v\|$, and $a$ is coercive (elliptic) :$a(u,u)\ge C_1\|u\|^2$, then there exists a bounded linear operator $A:X\to X$ with bounded inverse $A^{-1}$ satisfying $a(u,v)=\langle Au,v\rangle,\,\forall u,v\in X$ and $\|A\|\leq C_2$, $\|A^{-1}\|\leq \frac{1}{C_1}$ $\endgroup$ – Svetoslav Feb 16 '16 at 14:39
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    $\begingroup$ (2): Many linear elliptic equations can be brought in this form (via their weak formulation). Similarly, many nonlinear equations can be stated as $a(u;v) = b(v)$ with $a$ being nonlinear in the first argument. $\endgroup$ – gerw Feb 16 '16 at 14:39

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