  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 $\pi/6$, 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.
