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By da Prato G.

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Extra resources for Stochastic Partial Differential Equations

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1 jx log " C . 54) ˝" : Proof. " 1 x; y; log "/ defined in scaled coordinates in such a way that R. 66) R. 0; y/; R. ; y; log "/ ! 0 as j j ! 68) where y 2 ˝" : The solution of the above problem has the form R. 0; 0/g 1 C . 38). 74) is estimated by Const ", uniformly with respect to x 2 @F"c and y 2 ˝" . x; y/j Ä Const "; which is uniform for x; y 2 ˝" . 61)) also involves the capacitary potential P" from Sect. 1. 78) uniformly with respect to x; y 2 ˝" . 79) uniformly with respect to x; y 2 ˝" .

Jxj//. 90). t u Chapter 2 Mixed and Neumann Boundary Conditions for Domains with Small Holes and Inclusions: Uniform Asymptotics of Green’s Kernels In this chapter, we derive and justify asymptotic approximations of Green’s kernels for singularly perturbed domains whose boundary, or some part of it, supports the Neumann boundary condition. We also derive simpler asymptotic formulae, which become efficient when certain constraints are imposed on the independent variables. 2 deal with the Dirichlet–Neumann problems in twodimensional domains with small holes, inclusions or cracks.

42), implies for j j > 1 that 1 C j j jU. /j Ä C max jU. 42) once more, we complete the proof. 18). 2. The solution hN . 1 C jÁj/ 1 j j 2 j j2 ˇ ˇ ˇhN . 45) as j j > 2 and Á 2 R2 n F . Proof. The leading-order approximation of the harmonic function hN . ; Á/, as j j ! C1 1 C C2 2 /: Applying Green’s formula in BR n F to hN . ; Á/ and Dj . / limit, as R ! 1 jxjDR hN . Dj . Dj . / j/ Z D @F C. j Dj . // @hN . ; Á/ dS ; @n j; and taking the @hN . 46) where @=@n is the normal derivative in the direction of the inward normal with respect to F .

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