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Symmetric double bubbles in the Grushin plane. (English) Zbl 1439.53033

Symmetric double bubbles problem have been an active field over the past five decades. The authors in this paper consider and study the related double bubbles and obtain some interesting works. They address the double bubble problem for the anisotropic Grushin perimeter \(P_\alpha\), \(\alpha\geq 0\), and the Lebesgue measure in \(\mathbb{R}^2\), in the case of two equal volumes. By the hypothesis that the contact interface between the bubbles lies on either the vertical or the horizontal axis. The authors 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, they also prove that angles at which minimal boundaries intersect satisfy the standard 120-degree rule up to a suitable change of coordinates. While for \(\alpha=0\) the Grushin perimeter reduces to the Euclidean one and both minimizers coincide with the symmetric double bubble found in [J. Foisy et al., Pac. J. Math. 159, No. 1, 47–59 (1993; Zbl 0738.49023)], for \(\alpha=1\) 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 [R. Monti and D. Morbidelli, J. Geom. Anal. 14, No. 2, 355–368 (2004; Zbl 1076.53035)], in fact, the authors thought that the double bubble problem with no assumptions on the contact interface is solved by Monti and Morbidelli.

MSC:

53C17 Sub-Riemannian geometry
49Q20 Variational problems in a geometric measure-theoretic setting
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References:

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