Convexity of capillary surfaces in the outer space

Sharp size estimates for capillary free surfaces without gravity

“…For the convexity of the capillary free surface, in [2], Chen and Huang have shown if Ω is a bounded convex domain in the plane and θ o = 0, then the corresponding capillary surface is also convex. Finn [3] provided an example to show if θ o = 0 the result is in general false.…”

Uniqueness of the critical point of the solutions to some semilinear elliptic boundary value problems in 𝑅²

“…On the other hand, it follows from the result of Hartman and Wintner [8, Corollary 1, p. 450] that every interior critical point of u -w is isolated. Therefore, as in [5] we see that the zero set of u -w in some neighborhood U of the origin consists of n smooth arcs, all intersecting at origin and dividing U into 2n sectors (n 2: 3). Put…”

“…His results say that the solution u to (1.4) has only one critical point under the hypothesis of the existence of the solution with y = o. His proof is based on a nice comparison technique found in Chen and Huang [5] and the method of continuity with respect to y and the result of Chen and Huang [5] (that is, "the solution with y = 0 is strictly convex").…”

Superharmonicity of curvatures for surfaces of constant mean curvature

“…The superharmonicity of logK n proves the convexity of capillary surfaces during the deformation, valid for γ = 0 and arbitrary dimension n [Theorems 2 and 4]. This can be regarded as a generalization of the results by Chen-Huang [4] and Korevaar [11] where the nonparametric case was considered. Nonconvex examples of the problem were obtained by Finn [5] for nonzero γ.…”

“…For the case of tube with given cross section Ω where M is expressed nonparametrically in u = u(x), x £ Ω, related results have been obtained. In a joint work of the author with J. T. Chen [4], it was proved that for γ = 0 the convexity of Ω implies the strict convexity of the surface M in a strong sense that the Gaussian curvature is positive at the interior points. (In this paper we keep on using strictly convex to name the convexity in this strong sense.)…”