Water waves problem with surface tension in a corner domain I: A priori estimates with constrained contact angle

We study the two dimensional water waves problem with surface tension in the case when there is a non-zero contact angle between the free surface and the bottom. In the presence of surface tension, dissipations take place at the contact point. Moreover, when the contact angle is less than $\pi \mathrm{/}6$, no singularity appears in our settings. Using elliptic estimates in corner domains and a geometric approach, we prove an a priori estimate for the water waves problem.