I am frequently confused the definition of sup- and super-solutions so yesterday I tried to figure out a way to remember those things. Fortunately, I think I got a very simple way to remember. The point is, just denote by and the sup- and super-solutions respectively to the very simple PDE
which one is true
and similarly to .
Let us consider a general case. Assume we are working with a general second-order elliptic operator in non-divergence form, say
where are positive coefficients. Besides, coefficients of are assumed to satisfy several conditions like symmetry, etc. but it is not considered here.
Concerning the following PDE
function (resp. ) is said to be a super-solution (resp. sub-solution) to the PDE if (resp. ).
Observe that is positive lets us think that has positive spectrum. So once is a super-solution, the inequality should be where the left hand side should be an operator with positive spectrum. Similarly, concerning the sub-solutions, we need .
Let us go back to the Laplacian operator . So has negative spectrum, this yields
since has positive spectrum.