Ngô Quốc Anh

May 21, 2010

How to remember the definition of sup- and super-solutions?

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 17:10

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 \underline u and \overline u the sup- and super-solutions respectively to the very simple PDE

-\Delta u = f(x,u),

which one is true

\displaystyle -\Delta \overline u \leqslant f(x,\overline u)

or

\displaystyle -\Delta \overline u \geqslant f(x,\overline u)

and similarly to \underline u.

Let us consider a general case. Assume we are working with a general second-order elliptic operator in non-divergence form, say

\displaystyle L=-a^{ij}\partial_i\partial_j + b^k\partial_k + c

where a^{ij} are positive coefficients.  Besides, coefficients of L are assumed to satisfy several conditions like symmetry, etc. but it is not considered here.

Concerning the following PDE

L(u)=f(x,u),

we say

function \overline u (resp. \underline u) is said to be a super-solution (resp. sub-solution) to the PDE if L(\overline u) \geqslant f(x,\overline u) (resp. L(\underline u) \leqslant f(x,\underline u)).

Observe that ^{ij} is positive lets us think that L has positive spectrum. So once \overline u is a super-solution, the inequality should be {\rm LHS} \geqslant {\rm RHS} where the left hand side should be an operator with positive spectrum. Similarly, concerning the sub-solutions, we need {\rm LHS} \leqslant {\rm RHS}.

Let us go back to the Laplacian operator \Delta = \partial^i\partial_i. So \Delta has negative spectrum, this yields

\displaystyle \Delta \overline u \leqslant -f(x,\overline u)

and

\displaystyle \Delta \underline u \geqslant -f(x,\underline u).

Equivalently,

\displaystyle -\Delta \overline u \geqslant f(x,\overline u)

and

\displaystyle -\Delta \underline u \leqslant f(x,\underline u)

since -\Delta has positive spectrum.

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