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We study the two dimensional water waves problem with surface tension in thecase when there is a non-zero contact angle between the free surface and thebottom. In the presence of surface tension, dissipations take place at thecontact point. Moreover, when the contact angle is less than $\pi/6$, nosingularity appears in our settings. Using elliptic estimates in corner domainsand a geometric approach, we prove an a priori estimate for the water wavesproblem.

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