In this article we establish C3,α-regularity of the reduced boundary of stationary points of a nonlocal isoperimetric problem in a domain Ω⊂Rn. In particular, stationary points satisfy the corresponding Euler–Lagrange equation classically on the reduced boundary. Moreover, we show that the singular set has zero (n–1)-dimensional Hausdorff measure. This complements the results in [4] in which the Euler–Lagrange equation was derived under the assumption of C2-regularity of the topological boundary and the results in [27] in which the authors assume local minimality. In case Ω has non-empty boundary, we show that stationary points meet the boundary of Ω orthogonally in a weak sense, unless they have positive distance to it.
1
The New York Times Company, New York City, USA
2
Rocket Internet SE, Berlin, Germany
Dorian Goldman; Alexander Volkmann. On the regularity of stationary points of a nonlocal isoperimetric problem. Interfaces and free boundaries, Tome 17 (2015) no. 4, pp. 539-553. doi: 10.4171/ifb/353
@article{10_4171_ifb_353,
author = {Dorian Goldman and Alexander Volkmann},
title = {On the regularity of stationary points of a nonlocal isoperimetric problem},
journal = {Interfaces and free boundaries},
pages = {539--553},
year = {2015},
volume = {17},
number = {4},
doi = {10.4171/ifb/353},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/353/}
}
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AU - Dorian Goldman
AU - Alexander Volkmann
TI - On the regularity of stationary points of a nonlocal isoperimetric problem
JO - Interfaces and free boundaries
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EP - 553
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IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.4171/ifb/353/
DO - 10.4171/ifb/353
ID - 10_4171_ifb_353
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