, Conversely, if u s is a solution to problem (41), then the function v s which is equal to u s in ? R and S(u s | ?B R , (?u s /?n)| ?B R ) in R 2 \ B
, Assume that (1) has a solution v s in ?. It is clear that its restriction to R 2 \ B R coincides with S(v s | ?B R , (?v s /?n)| ?B R ), so that T(v s | ?B R , (?v s /?n)| ?B R ) = (N v s , M v s )
, To prove that v s is a solution to (1), we just have to verify that it satisfies ? 2 v s ? k 4 v s = 0 in ?. By construction, it satisfies this equation separately in ?
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