(Fredholm alternative) Suppose is a compact mapping of Hilbert space into itself. Then there exists a countable set having limit points except possible such that

(1)If , then

and

is uniquely solvable for any . And the inverse mappings , are bounded.

(2)If , the the has positive finite dimension and is solvable if and only if is orthogonal to the . And vice versa.

For a generalized second order elliptic equation

Usually we require is strictly elliptic, . Since the

then is invertible for big enough. So is equivalent to

Let , since is compact, then is compact. Then any satisfies the above equation is a weak solution in

with

In the general case to solve , this is equivalent to

So there exists a series of such that has nontrivial solutions. And the above equation is solvable if and only if

.

for any

For instance, consider the following equation

If

has no nontrivial solution, then is uniquely solvable.

Otherwise is solvable if and only if for any solves . The question is when has only the trivial solution.

This has much relation with the poincare inequality, since if satisfies , we will have ( must have some regularity to justify this).

For , we have

where is the volume of unit ball in .

If can be bounded by an n-dimensional strip with width , we have

So if is small enough, or is thin enough, will only have the trivial solution.

**Remark:** Why the is an operator on instead of the ? Pay attention to the “normal to the ” in fredholm alternative.