Boundary behavior of capillary surfaces possibly with extremal boundary angles

For solutions to the capillarity problem possibly with the boundary contact angle θ being 0 and/or π in a relatively open portion of the boundary which is C 2 , we will show that if the solution is locally bounded up to this portion of boundary, the trace of the solution on this portion is piecewise Lipschitz continuous and the solution is Hölder continuous up to the boundary, provided the prescribed mean curvature is bounded from above and from below. In the case where θ is not required to be bounded away from π / 2 , 0 , and π , and the mean curvature H ( x , t 0 ) belongs to L p ( Ω ) for some t 0 ∈ ℠and p > n , under the assumption that in a neighborhood of a relatively open portion of the boundary the solution is of rotational symmetry, the trace of the solution on this portion of the boundary is shown to be Hölder continuous with exponent 1 / n if n ≥ 3 and with exponent 1 / 3 if n = 2 .