41.1 The Case Of A Half Space
Regularity theorems are concerned with obtaining more regularity given a weak solution.
This extra regularity is essential in order to obtain error estimates for various
problems. In this section a regularity is given for weak solutions to various elliptic
boundary value problems. To save on notation, I will use the repeated index
summation convention. Thus you sum over repeated indices. Consider the following
Here V is an open set,
and U1 is an open set as shown for which U1 ⊆ V ∩ U. Assume also that V is bounded.
The following technical lemma gives the essential ideas.
Lemma 41.1.1 Suppose and
for all z ∈ H1
having the property that spt
⊆ V. Then w ∈ H2
and for some constant C, independent of f,w, and g, the following estimate
Proof: Define for small real h,
Let U1 ⊆U1 ⊆ W ⊆W ⊆ V and let η ∈ Cc∞
= 1 on U1
as shown in the following picture.
For h small (3h < dist
where here k < n
. Thus z
can be used in equation 41.1.7
. Begin by estimating the left
side of 41.1.7
Now consider these two terms. From 41.1.2,
Using the Lipschitz continuity of αrs and 41.1.12,
To see this, observe that if w is smooth, then
so by density of such functions in H1
holds. Therefore, changing ε,
With 41.1.14 and 41.1.18 established, consider the other terms of 41.1.7.
The following inequalities in 41.1.14