 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 satisfyingRepeated 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 $\varphi \in W_0^{1,2}(\Omega )$. Symmetry permits interchanging the derivative indices in the left side.