= Solution
The space $W_0^{1,2}(\Omega)$ is the closure of $C_c^\infty(\Omega)$ in the <Sobolev norm>; membership imposes the homogeneous Dirichlet condition in the weak sense, without assuming that the boundary is smooth. A <weak solution> is an element of that space satisfying
$$
\boxed{\int_\Omega a_{ij}D_i uD_j\phi\,dx=-\int_\Omega f\phi\,dx\quad\text{for every }\phi\in C_c^\infty(\Omega).}
$$
Repeated indices are summed. The minus sign comes from integration by parts in the divergence equation. Boundedness of the coefficients, <Hölder's inequality> and density extend this identity to every $\phi\in W_0^{1,2}(\Omega)$. Symmetry permits interchanging the derivative indices in the left side.
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