Title: Existence, Non-existence, and Regularity of Minimizers in Weighted Fractional Cluster ProblemsAbstract: Recently, fractional partitioning problems have attracted significant attention: given a domain Omega, how should it be divided to minimize a weighted fractional perimeter? These problems arise naturally as sharp-interface limits of fractional vectorial Allen-Cahn equations. Unlike the classical setting, where the weights correspond […]

Title: Fast Diffusion Equations on Bounded Domains. Sharp Extinction Rates and singular asymptotic behaviourAbstract: We begin with an overview of the ``classical'' homogeneous Dirichlet problem for the Fast Diffusion Equation $u_t=Delta u^m$, posed in a smooth bounded domain $Omegasubset mathbb{R}^N$, with $min (0,1)$.We then focus to the ``supercritical'' exponent range $m_s=(N-2)_+/(N+2)<m<1$ and focus on the […]