Polyharmonic capacity and Wiener test of higher order
In the present paper we establish the Wiener test for boundary regularity of the solutions to the polyharmonic operator. We introduce a new notion of polyharmonic capacity and demonstrate necessary and sufficient conditions on the capacity of the domain responsible for the regularity of a polyharmon...
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| Vydáno v: | Inventiones mathematicae Ročník 211; číslo 2; s. 779 - 853 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.02.2018
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| Témata: | |
| ISSN: | 0020-9910, 1432-1297, 1432-1297 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | In the present paper we establish the Wiener test for boundary regularity of the solutions to the polyharmonic operator. We introduce a new notion of polyharmonic capacity and demonstrate necessary and sufficient conditions on the capacity of the domain responsible for the regularity of a polyharmonic function near a boundary point. In the case of the Laplacian the test for regularity of a boundary point is the celebrated Wiener criterion of 1924. It was extended to the biharmonic case in dimension three by Mayboroda and Maz’ya (Invent Math 175(2):287–334,
2009
). As a preliminary stage of this work, in Mayboroda and Maz’ya (Invent Math 196(1):168,
2014
) we demonstrated boundedness of the appropriate derivatives of solutions to the polyharmonic problem in arbitrary domains, accompanied by sharp estimates on the Green function. The present work pioneers a new version of capacity and establishes the Wiener test in the full generality of the polyharmonic equation of arbitrary order. |
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| ISSN: | 0020-9910 1432-1297 1432-1297 |
| DOI: | 10.1007/s00222-017-0756-y |