Moon hypersurfaces and some related existence results of capillary hypersurfaces without gravity and of rotational symmetry

Absolute gradient bounds for nonparametric hypersurfaces of constant mean curvature and the structure of generalized solutions for the constant mean curvature equation

“…Finn therefore conjectured that the manifolds of solutuions in B R were to be progressively more restrictive as R → 1; for instance, there exist apriori es timates for solutions u(x ) to (0.1) in B R , which depend only on R and in no other way on the solutions u(x ), at least for R sufficiently close to 1. In this connection, we may recall the following results which have been proved in [6], [15] and [16] for n = 2, 3 ≤ n < 8 and n ≥ 8, respectively. (See also [10], Sect.…”

“…3.2 of [14] and Sect. 2.2 of [15], provided a proof for the intuitively obvious fact that, as ε → 0, | u ε | → ∞ at almost every point of Σ 1 (R); Σ 1 (R) being contacting ∂B R * , the Main Theorem II is seen to hold.…”

“…6, Sect. 9 of [15] for n ≥ 3. We also note that (0.3) is obtained by an integration of (0.4) over the cross-section Ω * and therefore is a necessary condition for the existence of the solution V r to the problem (0.4).…”

“…5, Sect. 9 of [15] and [16]; namely, there we have constructed the comparison (hyper)-surface V R , 1 > R > n−1 n , as the limit of solutions u ε to (0.1) defined throughout the sphere B R that is obtained by completing Σ 2 (R); furthermore, we have, in Sect. 3.2 of [14] and Sect.…”

“…In Fact, by recalling the Main Theorem I of [15] and normalizing the solutions u k by requiring u k (0) = 0 at the center 0 of B R , we have…”