A monotonicity formula for free boundary surfaces with respect to the unit ball

We prove a monotonicity identity for compact surfaces with free boundaries inside the boundary of unit ball in $\mathbb R^n$ that have square integrable mean curvature. As one consequence we obtain a Li-Yau type inequality in this setting, thereby generalizing results of Oliveira and Soret, and Fraser and Schoen. In the final section of this paper we derive some sharp geometric inequalities for compact surfaces with free boundaries inside arbitrary orientable support surfaces of class $C^2$. Furthermore, we obtain a sharp lower bound for the $L^1$-tangent-point energy of closed curves in $\mathbb R^3$ thereby answering a question raised by Strzelecki, Szuma\'nska and von der Mosel.