Boundary rectifiability and elliptic operators with W1,1 coefficients

Tatiana Toro; Zihui Zhao

doi:10.1515/acv-2017-0044

Published by De Gruyter October 17, 2017

Boundary rectifiability and elliptic operators with W^1,1 coefficients

Tatiana Toro and Zihui Zhao

From the journal Advances in Calculus of Variations

https://doi.org/10.1515/acv-2017-0044

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Abstract

We consider second-order divergence form elliptic operators with W1,1 coefficients, in a uniform domain Ω with Ahlfors regular boundary. We show that the A∞ property of the elliptic measure associated to any such operator implies that Ω is a set of locally finite perimeter whose boundary, ∂⁡Ω, is rectifiable. As a corollary we show that for this type of operators, absolute continuity of the surface measure with respect to the elliptic measure is enough to guarantee rectifiability of the boundary. In the case that the coefficients are continuous we obtain additional information about Ω.

Keywords: Elliptic measure; uniform domain; exterior corkscrew; rectifiability

MSC 2010: 35J25; 42B37; 31B35

Communicated by Jose Maria Martell

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1361823

Award Identifier / Grant number: DMS-1664867

Award Identifier / Grant number: DMS-1500098

Award Identifier / Grant number: DMS-1440140

Funding statement: The first author was partially supported by the Robert R. & Elaine F. Phelps Professorship in Mathematics, the Craig McKibben & Sarah Merner Professor in Mathematics and DMS grants 1361823 and 1664867. The second author was partially supported by DMS grants 1361823, 1500098 and 1664867. This material is based upon work supported by the National Science Foundation under DMS grant 1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.

Acknowledgements

We would like to thank the referee for the careful reading of the paper and very useful suggestions.

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Received: 2017-08-03

Revised: 2017-09-12

Accepted: 2017-09-14

Published Online: 2017-10-17

Published in Print: 2021-01-01

Boundary rectifiability and elliptic operators with W^1,1 coefficients

Abstract

Acknowledgements

References

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