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Regularity of solutions to the polyharmonic equation in general domains

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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.

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Notes

  1. The idea of the proof has been suggested by Marcel Filoche.

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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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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St SE, Minneapolis, MN, 55408, USA

    Svitlana Mayboroda

  2. Department of Mathematical Sciences, M&O Building, University of Liverpool, Liverpool, L69 3BX, UK

    Vladimir Maz’ya

  3. Department of Mathematics, Linköping University, 581 83, Linköping, Sweden

    Vladimir Maz’ya

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  1. Svitlana Mayboroda
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  2. Vladimir Maz’ya
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Correspondence to Svitlana Mayboroda.

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Mayboroda, S., Maz’ya, V. Regularity of solutions to the polyharmonic equation in general domains. Invent. math. 196, 1–68 (2014). https://doi.org/10.1007/s00222-013-0464-1

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