Sharp size estimates for capillary free surfaces without gravity

For the equation of constant mean curvature with prescribed constant contac t angle boundary condition, using the unique continuation of analytic funct ion, we get a minimum principle for a combination of the solution and its g radient. Thus we get the endpoint case for P-function (Sperb, 1981) and in fact answer an open question which appeared twenty years ago in Payne & Phi lippin, 1977, 1979 and Sperb, 1981. As an application, sharp size and shape estimetes for capillary free surface without gravity are obtained.