= Solution
Write $q_i=a^{ij}D_j u$. If $u$ is a <classical solution>, multiply $-\partial_iq_i=f$ by a smooth function $v$ and apply the <divergence theorem>. The <Neumann boundary condition> $q_iN_i=0$ removes the boundary term and gives
$$
\int_Ua^{ij}D_juD_iv
=\int_Ufv.
$$
The <density of smooth functions in a Sobolev space> and boundedness of the coefficients extend this identity to every $v\in H^1(U)$, so $u$ is a <weak solution>.
Conversely, take $v$ to be a <test function> compactly supported in $U$. The <weak formulation> says
$$
\int_U\bigl[-\partial_i(a^{ij}\partial_ju)-f\bigr]v=0.
$$
The <fundamental lemma of the calculus of variations> gives the equation in $U$. Under the regularity implicit in the stated notion of a classical solution, it holds pointwise. Applying <integration by parts> again with arbitrary $v\in H^1(U)$ leaves
$$
\int_{\partial U}(a^{ij}\partial_ju)N_i\,Tv=0,
$$
where $T$ is the <Sobolev trace theorem>[trace operator]. Traces of smooth functions can be chosen arbitrarily on the boundary, so the boundary <fundamental lemma of the calculus of variations> gives $(a^{ij}\partial_ju)N_i=0$. Thus $u$ is a classical solution.
Solved by gpt-5.6-sol high.
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