On the isoperimetric problem with respect to a mixed Euclidean-Gaussian density. (English) Zbl 1222.49058
Summary: The isoperimetric problem with respect to the product-type density \(e^{-\frac{|x|^2}{2}}dx\,dy\) on the Euclidean space \(\mathbb R^h \times \mathbb R^k\) is studied. In particular, existence, symmetry and regularity of minimizers is proved. In the special case \(k=1\), also the shape of all the minimizers is derived. Finally, a conjecture about the minimality of large cylinders in the case \(k>1\) is formulated.
MSC:
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 49N60 | Regularity of solutions in optimal control |
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