Theorem 1 Let with , such that , where , if ; for all , if . If satisfies
Here is a famous fixed point theorem in finite dimension by Brouwer:
Theorem 1 (Brouwer fixed point theorem) Let be a convex compact set, for any continuous function , there exists a point such that .
There are couple of ways to extend this theorem to Banach spaces. First recall that a mapping between two Banach spaces is called compact, if the mapping is continuous (not necessarily linear) and the images of bounded sets are pre-compact.
Definition 1 Let . Define
Then is called d-dimensional Hausdorff measure in .
Remark 1 Hausdorff measure is a Borel outer measure, it is not Radon for because is not finite with respect to . And it is a measure if restricted on Lebesgue measurable sets (by Caratheodory condition: We say a set satisfies Caratheodory condition, if for any set , ). Moreover, note that gives the volume of unit ball in dimension if is an integer, we naturally have (not trivially) , where is the n-dimensional Lebesgue measure.
We will show a basic argument why Harnack inequality implies Holder continuity. Assume is a weak solution to a uniform elliptic equation
in . Assume we have Harnack inequality (Indeed we do have due to Moser), i.e. for any positive weak solution we will have
where is a constant independent of .
First one will know , see this note.
If is a weak solution, so are and . And they are strictly positive in due to the strong maximum principle, otherwise will be a constant. By Harnack inequality, we will have
Adding these two inequalities up we will have
Denote , and choose such that , by interation we know
Then for any , choose such that , we will have
So Holder continuity follows.
— 1. Definition of Newtonian Potential —
Definition 1 Let be a bounded open set, we define the Newtonian potential of is the function on by
where is the fundamental solution of Laplace’s equation.
Theorem (Analytic Fredholm Alternative) Let be a connected open subset of and be a separable Hilbert space. Suppose that is an analytic operator-valued function such that is compact . Then either
- does not exist .
- exists , where is discrete.
Theorem Suppose is a subsolution of , . Then for all , ,