Let us start our notes with a very fundamental maximum principle for any strongly second order elliptic operator. We have

Theorem(Maximum principle). Let satisfy the differential inequalityin a domain where is uniformly elliptic. Suppose the coefficients and are uniformly bounded. If attains a maximum at a point of , then in .

In order to memorize the above result, let us think about the parabola with and . In this one-dimentional case, which confirms that only achieves its maximum at .

As you may know the operator is only assumed to be strongly elliptic which only effects the coefficients . Regarding to the coefficients , we only assume these are uniformly bounded. However, the uniform ellipticity of the operator L and the boundedness of the coefficients are not really essential as you can check in the proof. Besides, the domain need not be bounded in this version.

Now for operators of the form , we still have a result analogous to the above.

Theorem(Maximum principle). Let satisfy the differential inequalitywith , with uniformly elliptic in , and with the coefficients of and bounded. If attains a non-negative maximum at an interior point of , then .

Clearly, the assumption and are crucial as we may face some difficulty as raised in this note. Counterexamples are easily obtained if . For example, the function has an absolute maximum at .