Volume 25, 2019
|Number of page(s)||37|
|Published online||05 December 2019|
Symmetric double bubbles in the Grushin plane
FMJH & IMO, Bâtiment 307, Faculté des Sciences d’Orsay, Université Paris Sud,
2 Scuola Normale Superiore, Piazza Cavalieri 7, 56126 Pisa, Italy.
* Corresponding author: email@example.com
Accepted: 9 October 2018
We address the double bubble problem for the anisotropic Grushin perimeter Pα, α ≥ 0, and the Lebesgue measure in ℝ2, in the case of two equal volumes. We assume that the contact interface between the bubbles lies on either the vertical or the horizontal axis. We first prove existence of minimizers via the direct method by symmetrization arguments and then characterize them in terms of the given area by first variation techniques. Even though no regularity theory is available in this setting, we prove that angles at which minimal boundaries intersect satisfy the standard 120-degree rule up to a suitable change of coordinates. While for α = 0 the Grushin perimeter reduces to the Euclidean one and both minimizers coincide with the symmetric double bubble found in Foisy et al. [Pacific J. Math. 159 (1993) 47–59], for α = 1 vertical interface minimizers have Grushin perimeter strictly greater than horizontal interface minimizers. As the latter ones are obtained by translating and dilating the Grushin isoperimetric set found in Monti and Morbidelli [J. Geom. Anal. 14 (2004) 355–368], we conjecture that they solve the double bubble problem with no assumptions on the contact interface.
Mathematics Subject Classification: 53C17 / 49Q20
Key words: Calculus of variations / shape optimization / sub-Riemannian manifold / Grushin perimeter / double bubble problem / constrained interface / 120-degree rule
© The authors. Published by EDP Sciences, SMAI 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.