We prove a pointwise control for the Green's function of polyharmonic
operators with holes: this control is uniform while holes shrink. For the usual
Laplacian, such a control is given by the maximum principle; the techniques
developed here applies to general polyharmonic operators for which there is no
comparison principle

Grunau, Hans-Christoph

Robert, Frédéric

English

arXiv

Dedicated to Patrizia Pucci on the occasion of her 60th birthdayInternational audienceThe Green function G −∆,Ω for the Laplacian under Dirichlet boundary conditions in a bounded smooth domain Ω ⊂ R n enjoys in dimensions n ≥ 3 the estimate: 0 ≤ G −∆,Ω (x, y) ≤ 1 n(n − 2)en |x − y| 2−n. Here, en denotes the volume of the unit ball B = B 1 (0) ⊂ R n. This estimate follows from the maximum principle, the construction of G −∆,Ω and the explicit expression of a suitable fundamental solution. When passing to the polyharmonic Green function G (−∆) k ,Ω under Dirich-let boundary conditions almost all forms of maximum or comparison principles fail: Green function estimates become an intricate subject and, according to works of Krasovski˘ ı, multiplicative constants have to be used which heavily depend on the smoothness properties of the underlying domains. In the present paper we study a singular family of domains by removing arbitrarily small holes from a fixed smooth domain in R n with n > 2k. We prove Green function estimates which are uniform even when the size of the hole approaches 0, i.e. when the curvature of the boundary becomes unbounded

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Uniform estimates for polyharmonic Green functions in domains with small holes

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Uniform estimates for polyharmonic Green functions in domains with small holes

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