  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.
