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 the unit ball in 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 [19, Proposition 3], and Fraser and Schoen [11, Theorem 5.4]. 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 R-3 thereby answering a question raised by Strzelecki, Szumanska, and von der Mosel [22].