Entropy flux - electrostatic capacity - graphical mass

Abstract:

This note shows that the optimal inequality \[ \mathsf {F}(K,\kappa )\le \mathsf {C}(K)\le 2(n-2)\sigma _{n-1}\mathsf {M}(\mathbb R^n\setminus K^\circ ,\delta +df\otimes df) \] holds for the entropy flux $\mathsf {F}(K,\kappa )$, the electrostatic capacity $\mathsf {C}(K)=\mathsf {C}(\partial K)$ and the graphical mass $\mathsf {M}(\mathbb R^n\setminus K^\circ ,\delta +df\otimes df)$ generated by a compact $K\subset \mathbb R^{n\ge 3}$ with nonempty interior $K^\circ$ and smooth boundary $\partial K$.