Here we prove that if u satisfies the minimal surface equation in an unbounded domain which is properly contained in a half space of R n , with n ≥ 2, then the growth rate of u is of the same order as that of the shape of Ω and the boundary value of u.
In Part 1 of this work, we shall resort to the method of interior parallels and give a proof of an isoperimetric inequality of the type of Bol-Fiala-Huber for a multiply-connected, compact surface with boundary whose Gauss curvature is bounded above by a number K 0. Also, we shall estimate the upper bound of the radius of the largest circle inscribed in such a surface in terms of its area and K 0. In Section 1.3, we shall improve this estimate by introducing a quantity − * (ρ), which depends on the perimeter of M , the area of M and also the shape of M. In terms of this quantity, together with the perimeter of M , area of M , and the lower bound of the Gauss curvature of M , we estimate the lower bound of the largest radius of inscribed circle in Section 1.4. In Part 2, using the results in Part 1, we obtain some estimates in the torsion problem. In Part 3, we use the warping function as the trial function in the Rayleigh quotient for the first eigenvalue λ 1 of the fixed membranes in M , and obtain upper bound of λ 1 in terms of the area of M , the perimeter of M , together with the lower and upper bounds of the Gauss curvature of M .
We obtain global estimates for the modulus, interior gradient estimates, and boundary Hölder continuity estimates for solutions u to the capillarity problem and to the Dirichlet problem for the mean curvature equation merely in terms of the mean curvature, together with the boundary contact angle in the capillarity problem and the boundary values in the Dirichlet problem.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.