Uniformly elliptic divergence-form operators and their sesquilinear forms

Assume Countable Choice for the Sobolev interfaces. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open, $n\ge 1$, and let $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. Let $a^{ij},b^i,c:\mathrm{\Omega }\to \mathbb{K}$, $i,j=1,\dots ,n$, be measurable (A measurable function between measurable spaces) and essentially bounded (The space $L^{\mathrm{\infty }}(\mu )$ of essentially bounded measurable functions, The essential supremum of a measurable function with respect to a measure), with bounds $\mathrm{\mid }{a}^{ij}\mathrm{\mid }\le M_a,\mathrm{\mid }{b}^i\mathrm{\mid }\le M_b,\mathrm{\mid }c\mathrm{\mid }\le M_c\text{a.e. on }\mathrm{\Omega },$ and suppose the uniform ellipticity condition holds: there is $\theta >0$ with $\mathrm{Re}({\sum }_{i,j=1}^n{a}^{ij}(x){\xi }_j\overline{{\xi }_i})\ge \theta \mathrm{\mid }\xi {\mathrm{\mid }}^2\text{for a.e. }x\in \mathrm{\Omega }\text{ and all }\xi \in {\mathbb{C}}^n\mathrm{.}$ The associated divergence-form expression is $Lu:=-D_i(a^{ij}{D}_{j}u)+b^i{D}_{i}u+cu$ (Einstein summation over $i,j$), and the associated sesquilinear form on $H^1(\mathrm{\Omega })$ is $a(u,v):={\int }_{\mathrm{\Omega }}(a^{ij}{D}_{j}u\overline{D_{i}v}+b^i{D}_{i}u\bar{v}+cu\bar{v})dx\mathrm{.}$ The form is linear in $u$ and conjugate-linear in $v$, in the convention of Bounded, coercive and symmetric sesquilinear forms; the classical Dirichlet problem consists of $Lu=f$ in $\mathrm{\Omega }$ with prescribed boundary values. The operator is determined by the coefficient functions only a.e., and all later statements about $a$ are statements about those classes. Where the domain and boundary data demand it (Weak Dirichlet solutions for a divergence-form operator, The inhomogeneous weak Dirichlet problem by a trace lifting) the domain is additionally a bounded $C^1$ domain (Bounded C^k domains and boundary charts). The convention is the complex sesquilinear one with conjugation in the second slot, as the plan's convention audit directs; the real case is the same with conjugation read as the identity.