16:15 Uhr | Aldo Pratelli (Univ. Erlangen) | The isoperimetric problem in a space with density: existence,
boundedness and regularity of solutions.
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).
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17:45 Uhr | Andreas Savas-Halilaj (Hannover) | Bernstein type theorems in higher co-dimensions
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.
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