On the isoperimetric problem with respect to a mixed Euclidean–Gaussian density

The isoperimetric problem with respect to the product-type density e −|x|2 2 dx dy on the Euclidean space Rh ×Rk 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. © 2011 Elsevier Inc. All rights reserved