We will discuss the isoperimetric problem in $\mathbb R^N$ with a density; that is, one wants to minimize the perimeter of sets with fixed volume, as usual for the isoperimetric problem, but "perimeter" and "volume" are considered with respect to a given l.s.c. density. Many particular cases of this problem have been and are presently well studied in view of their importance in applications, while only very few results were known under general assumptions on the density. In particular, almost nothing until very recently was known in the case of densities which are not at least Lipschitz. We will concentrate on the very general case of densities which are just Hölder continuous, or even just continuous, and we will study the three main properties: existence of isoperimetric sets, boundedness, and regularity. We will also discuss several open questions. (Joint works with E. Cinti and with F. Morgan).

Based on works by Hopf, Weinberger, Hamilton and Evans, we derive a strong elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds and use it to prove Bernstein type theorems in higher co-dimensions for minimal maps between Riemannian manifolds. This is joint work with Knut Smoczyk.