Chapter 41

Elliptic Regularity41.1 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 valueproblems. To save on notation, I will use the repeated index summation convention. Thusyou sum over repeated indices. Consider the following picture.

R

Rn−1

U

V

Γ

U1

Here V is an open set,

U ≡ {y ∈V : yn < 0} ,Γ≡ {y ∈V : yn = 0}

and U1 is an open set as shown for which U1 ⊆ V ∩U. Assume also that V is bounded.Suppose

f ∈ L2 (U) ,

αrs ∈C0,1 (U) , (41.1.1)

αrs (y)vrvs ≥ δ |v|2 , δ > 0. (41.1.2)

The following technical lemma gives the essential ideas.

Lemma 41.1.1 Suppose

w ∈ H1 (U) , (41.1.3)α

rs ∈ C0,1 (U) , (41.1.4)

hs ∈ H1 (U) , (41.1.5)f ∈ L2 (U) . (41.1.6)

and ∫U

αrs (y)

∂w∂yr

∂ z∂ys dy+

∫U

hs (y)∂ z∂ys dy =

∫U

f zdy (41.1.7)

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