The $L^2$ operator associated with a symmetric elliptic form

Assume Countable Choice. Symmetric case. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open and let $a$ be the divergence-form sesquilinear form of Uniformly elliptic divergence-form operators and their sesquilinear forms with $b^i\equiv 0$, coefficients satisfying $a^{ij}=\overline{a^{ji}}$ a.e. and real $c$, all measurable and essentially bounded, and with uniform ellipticity constant $\theta$. Thus $a(u,v)={\int }_{\mathrm{\Omega }}(a^{ij}{D}_{j}u\overline{D_{i}v}+cu\bar{v})dx$ is a bounded symmetric form, $a(u,v)=\overline{a(v,u)}$ (Bounded, coercive and symmetric sesquilinear forms, The formal adjoint and the adjoint weak Dirichlet problem). Define $D(L):=\{u\in H_0^1(\mathrm{\Omega }):\mathrm{\exists }f\in L^2(\mathrm{\Omega })\text{ with }a(u,v)=(f,v{)}_{L^2}\mathrm{\forall }v\in H_0^1(\mathrm{\Omega })\},Lu:=f\mathrm{.}$ This is well defined: if $f,g$ both satisfy the defining identity then $(f-g,v{)}_{L^2}=0$ for every $v\in H_0^1(\mathrm{\Omega })$, and $H_0^1(\mathrm{\Omega })$ is dense in $L^2(\mathrm{\Omega })$ (Smooth compactly supported functions of an open set are dense in $L^2$), so $f=g$ in $L^2(\mathrm{\Omega })$. The space $D(L)$ is a linear subspace of $H_0^1(\mathrm{\Omega })$ containing the range of every shifted solution operator (The shifted elliptic solution operator), and $L:D(L)\to L^2(\mathrm{\Omega })$ is linear. With only bounded measurable coefficient hypotheses, $D(L)$ may be a proper subspace of the form domain $H_0^1(\mathrm{\Omega })$; those hypotheses alone do not assert $C_c^{\mathrm{\infty }}(\mathrm{\Omega })\subseteq D(L)$. Membership $u\in D(L)$ with $Lu=f$ is exactly the weak statement of $Lu=f$ with zero boundary values in $L^2$ data (Weak Dirichlet solutions for a divergence-form operator).

Well-definedness and symmetry, recorded with the definition. Boundedness of $a$ on $H^1(\mathrm{\Omega })$ is The elliptic form is well defined and bounded on $H^1$ with $b=0$, and symmetry follows by conjugating the defining integrand: with $a^{ij}=\overline{a^{ji}}$ and $c$ real, $\overline{a(v,u)}={\int }_{\mathrm{\Omega }}(\overline{a^{ji}}{D}_{j}u\overline{D_{i}v}+\bar{c}\bar{v}u)dx=a(u,v)$ after re-indexing. Hence the pair $a,⟨\cdot ,\cdot {⟩}_{L^2}$ is the symmetric sesquilinear pair whose weak identity defines $D(L)$. The representing datum is unique by the density argument above, so $Lu$ is a well-defined class; linearity of $L$ follows from linearity of $a$ and of the $L^2$ pairing. The range inclusion $\mathrm{ran}{K}_{\mu }\subseteq D(L)$ holds because $K_{\mu }g$ satisfies $a(K_{\mu }g,v)=a_{\mu }(K_{\mu }g,v)-\mu (K_{\mu }g,v{)}_{L^2}=(g-\mu K_{\mu }g,v{)}_{L^2}$ for all $v\in H_0^1(\mathrm{\Omega })$, with datum $g-\mu K_{\mu }g\in L^2(\mathrm{\Omega })$ (The shifted elliptic solution operator, The space $L^p(\mu )$ as the quotient by null functions, Integer-order Sobolev spaces and their norms, Zero-boundary Sobolev space as a norm closure, Real and imaginary parts, complex conjugation, and modulus, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). No claim of self-adjointness, closedness, density of $D(L)$, or identification with a classical differential expression is made here; those belong to the following items.