The elliptic form is well defined and bounded on $H^1$

Every integrand is measurable and bounded a.e. by a product of $L^2$ classes: the products $a^{ij}{D}_{j}u\overline{D_{i}v}$, $b^i{D}_{i}u\bar{v}$ and $cu\bar{v}$ are measurable by [F1] and [F3], since products of measurable functions are measurable and representatives agree a.e.; the a.e. coefficient bounds turn each of them into an a.e. dominated multiple of a product of two $L^2$ classes, e.g. $\mathrm{\mid }{a}^{ij}{D}_{j}u\overline{D_{i}v}\mathrm{\mid }\le M_a\mathrm{\mid }{D}_{j}u\mathrm{\mid }\mathrm{\mid }{D}_{i}v\mathrm{\mid }$ off a null set.

Principal part: for almost every $x$, Cauchy--Schwarz in ${\mathbb{R}}^n$ applied to the vectors $(\mathrm{\mid }{D}_{j}u(x)\mathrm{\mid }{)}_j$ and $(1,\dots ,1)$, together with $\mathrm{\mid }{a}^{ij}(x)\mathrm{\mid }\le M_a$, gives $\mathrm{\mid }{a}^{ij}(x)D_{j}u(x)\overline{D_{i}v(x)}\mathrm{\mid }\le n{M}_a\mathrm{\mid }Du(x)\mathrm{\mid }\mathrm{\mid }Dv(x)\mathrm{\mid },$ where $\mathrm{\mid }Du\mathrm{\mid }=({\sum }_j\mathrm{\mid }{D}_{j}u{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$. Hence ${\int }_{\mathrm{\Omega }}\mathrm{\mid }{a}^{ij}{D}_{j}u\overline{D_{i}v}\mathrm{\mid }dx\le n{M}_a\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^2}\mathrm{\parallel }Dv{\mathrm{\parallel }}_{L^2}$ by H"older with exponent $2$, so the principal term converges absolutely.

Drift part: summing the coefficientwise bounds and applying H"older to each $\mathrm{\mid }{D}_{i}u\mathrm{\mid }\mathrm{\mid }v\mathrm{\mid }$ gives ${\sum }_{i=1}^n{\int }_{\mathrm{\Omega }}\mathrm{\mid }{b}^i{D}_{i}u\bar{v}\mathrm{\mid }dx\le M_b{\sum }_{i=1}^n\mathrm{\parallel }{D}_{i}u{\mathrm{\parallel }}_{L^2}\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2}\le n{M}_b\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^2}\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2}$, since $\mathrm{\parallel }{D}_{i}u{\mathrm{\parallel }}_{L^2}\le \mathrm{\parallel }Du{\mathrm{\parallel }}_{L^2}$; the drift term is absolutely convergent.

Reaction part: ${\int }_{\mathrm{\Omega }}\mathrm{\mid }cu\bar{v}\mathrm{\mid }dx\le M_c\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2}$ by H"older.

Bound: adding the three estimates and using $\mathrm{\parallel }Dw{\mathrm{\parallel }}_{L^2}\le \mathrm{\parallel }w{\mathrm{\parallel }}_{H^1}$ and $\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2}\le \mathrm{\parallel }w{\mathrm{\parallel }}_{H^1}$ for $w=u$ and $w=v$ gives $\mathrm{\mid }a(u,v)\mathrm{\mid }\le n{M}_a\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}+n{M}_b\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}+M_c\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}=(n{M}_a+n{M}_b+M_c)\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}$, so $a$ is a bounded form on $H^1(\mathrm{\Omega })$.

Class independence and sesquilinearity: replacing $u$, $v$ or any coefficient by another representative alters each integrand only on a null set, hence leaves every integral unchanged; in particular the two $L^2$ and weak-derivative slots depend only on the classes, and the value is finite by steps 1.2--2.1. Linearity in $u$ and conjugate-linearity in $v$ hold termwise: the principal and drift terms are linear in the $u$-slot and conjugate-linear in the $v$-slot, and the reaction term is linear in $u$ and conjugate-linear in $v$, with the finite sum of absolutely convergent integrals linear separately in each slot. So $a$ is a well-defined bounded sesquilinear form on $H^1(\mathrm{\Omega })$; on the subspace $H_0^1(\mathrm{\Omega })$ the same estimate holds with the restricted norm.