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

Abstract

The isoperimetric problem with respect to the product-type density e − | x | 2 2 d x d y on the Euclidean space R h × 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.