Weak subsolution of a divergence-form elliptic equation ($Lu\ge 0$)

For the sign convention $Lu=D_i(a^{ij}{D}_{j}u)$, a weak subsolution satisfies $Lu\ge 0$ as a distribution, equivalently $\int a^{ij}{D}_{j}u{D}_i\varphi \le 0$ for every nonnegative compactly supported test function $\varphi$. A weak supersolution has the reversed inequality.