Monotonicity formulas for capillary surfaces

In this paper, we establish monotonicity formulas for capillary surfaces in the half-space $R_{+}^3$ and in the unit ball $B^3$ and extend the result of Volkmann (2016) [27] for surfaces with free boundary. As applications, we obtain Li-Yau-type inequalities for the Willmore energy of capillary surfaces, and extend Fraser-Schoen's optimal area estimate for minimal free boundary surfaces in $B^3$ (2011) [10] to the capillary setting, which is different to another optimal area estimate proved by Brendle (2023) [5].