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Uniform estimates for polyharmonic Green functions in domains with small holes

by Grunau, Hans-Christoph, Robert, Frederic.

Series: 2012-12, Preprints

35J40 Boundary value problems for higher-order elliptic equations
35B50 Maximum principles

We show that in domains in R^n with arbitrarily small smooth holes, the k-polyharmonic Green function (n>2k) can be estimated by C |x-y|^{2k-n}, i.e. a multiple of the polyharmonic singular solution. The key point is that the constant can be chosen independently of the size of the holes. According to more classical results one would have
expected this constant to blow up as the holes shrink to a point.

polyharmonic Green function, small holes, uniform estimates