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
or
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
,
we say
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
and
.
Equivalently,
and
since has positive spectrum.