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Unformatted text preview: 1 Lecture 9 Weakly singular integral operators Reference: Multidimensional Weakly Singular Integral Equations, Gennadi Vainikko, Lecture Notes in Mathematics, Vol. 1549, Springer-Verlag Some definitions: Let G ⊂ R n be an open bounded set. Then G consists of a finite or countable number of connectivity components (maximal connected open subsets of G ). For x,y in the same connectivity component of G , define d G ( x,y ) to be the infimum of the lengths of the polygonal paths in G joining x and y . For x,y in different connectivity components of G , let d G ( x,y ) = d * , where d * is the sup of d G ( x 1 ,x 2 ) as x 1 and x 2 vary in common connectivity components. Let ¯ G denote the closure of G and G * denote the completion of G with respect to the metric d G . Let BC( G * ) be the space of bounded continuous functions on G * . Note that k u k BC ( G * ) = sup x ∈ G * | u ( x ) | = sup x ∈ G | u ( x ) | = k u k L ∞ ( G ) Let A be the integral operator ( Au ) x = Z G K ( x,y ) dy Theorem 1 Suppose | K ( x,y ) | ≤ b | x-...
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This note was uploaded on 03/14/2011 for the course MATH 676 taught by Professor Staff during the Fall '08 term at Colorado State.
- Fall '08