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.

For a uniformly elliptic divergence-form elliptic operator with lower-order coefficients in the supercritical Lebesgue spaces, truncation tests and Moser iteration bound the positive part of a weak subsolution on an interior ball by its Lp norm on a larger ball and the forcing norms. In particular, every weak subsolution of the homogeneous equation is locally bounded above. This is the preliminary estimate needed when a strong maximum principle is phrased using essential suprema.