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Inventiones mathematicae

, Volume 196, Issue 1, pp 1–68 | Cite as

Regularity of solutions to the polyharmonic equation in general domains

  • Svitlana MayborodaEmail author
  • Vladimir Maz’ya
Article

Abstract

The present paper establishes boundedness of \([m-\frac{n}{2}+\frac{1}{2} ]\) derivatives for the solutions to the polyharmonic equation of order 2m in arbitrary bounded open sets of \({\mathbb{R}}^{n}\), 2≤n≤2m+1, without any restrictions on the geometry of the underlying domain. It is shown that this result is sharp and cannot be improved in general domains. Moreover, it is accompanied by sharp estimates on the polyharmonic Green function.

Keywords

Dirichlet Problem Fundamental Solution Lipschitz Domain General Domain Integral Identity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We are greatly indebted to Marcel Filoche for the idea relating certain positivity properties of one-dimensional differential operators to particular configurations of the roots of associated polynomials. It has been reflected in Sect. 3 and it has ultimately significantly influenced our technique.

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of Mathematical Sciences, M&O BuildingUniversity of LiverpoolLiverpoolUK
  3. 3.Department of MathematicsLinköping UniversityLinköpingSweden

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