This is a question originating in another mathematics forum, matematicamente.it (in Italian).
In literature one encounters the word elliptic in (at least) two different definitions. In what follows $\Omega$ is an open bounded subset of $\mathbb{R}^n$ and $\mathcal{D}(\Omega)$ denotes the space of smooth functions with compact support.
Definition 1 A differential operator (in divergence form)
$L(u)(x)=-\mathrm{div} \big( A(x)Du(x) \big) u(x), \qquad x \in \Omega$
is said to be (uniformly) elliptic (1) if there exists a $\theta >0 $ s.t. the matrix-valued function $A$ verifies
$A(x)\xi \cdot \xi \ge \theta \lvert \xi \rvert^2, \qquad x, \xi \in \mathbb{R}^n.$
Definition 2 A (densely defined) linear operator $(L, D(L))$ on a Hilbert space $H$ is said to be $H$-elliptic (2) if there exists a $c >0$ s.t.
$(Lu, u) \ge c \lVert u \rVert^2, \qquad u \in D(L).$
Question Let
$L(u)(x)=-\mathrm{div} \big( A(x)Du(x) \big) u(x), \quad D(L)=\mathcal{D}(\Omega),\quad H=L^2(\Omega).$
Is it true that $L$ is elliptic as in definition 1 if and only if it is $H$-elliptic as in definition 2? Assume that $A$ depends continuously on $x$ and is symmetric everywhere.
It is straightforward to prove that definition 1 implies definition 2; I find it nontrivial to prove the converse (if true).
What do you think?
1) cfr. Evans, Partial differential equations, §6.1.1.
2) cfr. Kesavan, Topics in functional analysis, §3.1.1.