Seminario di analisi matematica
ore
16:00
presso Seminario II
An old conjecture of Almgren states that for every convex and coercive
potential $g: \mathbb{R}^d\to \mathbb{R}$, every convex and
one-homogeneous anisotropy $\Phi : \mathbb{R}^d\to \mathbb{R}^+$
and every volume $V>0$, the minimizers of
\[
\min_{|E|=V} \int_{\partial E} \Phi(\nu) d\mathcal{H}^{d-1} + \int_{E}
g dx
\]
are convex. I will review the known results on this problem and present
recent progress obtained with G. De Philippis on the connectedness of
the minimizers for smooth potentials
and anisotropies. Our proof is based on the introduction of a new
``two-point function'' which measures the lack of convexity and which
gives rise to a negative second variation of the energy.