= Solution
Write $H^1=W^{1,2}$ and $H_0^1=W_0^{1,2}$. The <Dirichlet boundary condition> in the <Sobolev space> sense means $u-\psi\in H_0^1(\Omega)$; this definition remains meaningful even without regularity assumptions on the boundary. For bounded full coefficients, a <weak solution> is a function satisfying
$$
\boxed{u\in H^1(\Omega),\quad u-\psi\in H_0^1(\Omega),\quad B(u,\varphi)=-\int_\Omega f\varphi\quad\text{for every }\varphi\in H_0^1(\Omega),}
$$
where
$$
B(u,\varphi)=\int_\Omega a_{ij}D_ju\,D_i\varphi-\int_\Omega qu\varphi.
$$
The sign follows by <integration by parts> in $D_i(a_{ij}D_ju)+qu=f$. No classical derivatives of the measurable coefficients are taken. Equivalently one may first test against $C_c^\infty(\Omega)$ and extend by density, provided this <bilinear form> is bounded on $H^1\times H_0^1$. All integrals in this definition must exist: as explained below, the printed quadratic bounds alone do not guarantee this when $A=(a_{ij})$ is nonsymmetric.
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