### 2012-12

by Grunau, Hans-Christoph, Robert, Frederic.

**Series:** 2012-12, Preprints

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

**Abstract:**

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.

**Keywords:**

polyharmonic Green function, small holes, uniform estimates