By Lucia M.

We ponder the equationIf Ω is of sophistication , we convey that this challenge has a non-trivial answer u λ for every λ ∊ (8π, λ*). the worth λ* relies on the area and is bounded from lower than through 2 j zero 2 π, the place j zero is the 1st 0 of the Bessel functionality of the 1st type of order 0 (λ*≥ 2 j zero 2 π > eight π). additionally, the kin of resolution u λ blows-up as λ → eight π.

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1) The first IT is included as it is sometimes convenient to think of Tf : L2{Sl) -* H2^1). One can easily prove the estimate \\Tfu\\L2<\\f\\L~\\u\\L2. (2) This estimate can be read in two different ways. As an operator acting on u, Tf is a bounded operator with ||T/||Op < | | / | | i ~ . Since Tf ~Tg = Ty_9 this implies that P>-r9||op<||/-0||L~ 00 1 (3) OO provided f,g