Theorem 1Let with , such that , where , if ; for all , if . If satisfiesthen .

# Two fixed point theorems in Banach space

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.

# Hausdorff Measure and Capacity

Definition 1Let . Definewhere .

Then is called d-dimensional Hausdorff measure in .

Remark 1Hausdorff 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.

# Harnack inequality implies Holder continuity

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

hence

Denote , and choose such that , by interation we know

Then for any , choose such that , we will have

So Holder continuity follows.

# Newtonian Potential

**— 1. Definition of Newtonian Potential —**

Definition 1Let be a bounded open set, we define the Newtonian potential of is the function on bywhere is the fundamental solution of Laplace’s equation.

# Fredholm Alternative and Riesz Schauder Theorem

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.

# Regularity of scalar elliptic equation by Moser iteration

TheoremSuppose is a subsolution of , . Then for all , ,