content_image/306616.jpg content_image/306617.jpg content_image/306618.jpg content_image/306619.jpg content_image/306620.jpg content_image/306621.jpg content_image/306622.jpg
for all large s > 0. Now, we absorb the first term on the right-hand side into the left-hand side by choosing s > 0 large, we get 

for all large s > 0. 

Replacing the integration domain on the left-hand side by D × { y ; | y − y$_{0}$ | < L ρ } ⊂ Ω$_{0}$ and using the facts that χ$_{0}$ ( | y − y$_{0}$ | ) = 1 in D × { y ; | y − y$_{0}$ | < L ρ } and 

we obtain, 

<img src='content_image/306616.jpg'>

<img src='content_image/306617.jpg'>

<img src='content_image/306618.jpg'>

for all s ≥ s$_{0}$ , where s$_{0}$ is some constant. Since κ$_{2}$ > κ$_{1}$, the last inequality implies 

<img src='content_image/306619.jpg'>

for all s ≥ s$_{0}$ , where κ = κ$_{2}$ − κ$_{1}$ > 0. We separately consider the two cases: 

Case 1. Let M ≥ d. Choosing s ≥ 0 such that 

we obtain 

Case 2. Let M < d. Then setting s = 0 in (4.23) we have 

<img src='content_image/306620.jpg'>

<img src='content_image/306621.jpg'>

<img src='content_image/306622.jpg'>

