This work concerns boundary
value problems for a class of nonlinear equations modeled on the physical equations
for a capillary free surface in a gravitational field. The results consist principally of
estimates for the height of a solution in an exterior domain. Structure conditions
reflecting the nonlinearity of the mean curvature operator are imposed on a class of
symmetric variational operators in terms of the Legendre transform of the variational
integrand. Estimates are found for the boundary height of a rotationally symmetric
solution in the exterior of a ball of radius R. These estimates, which are valid for
any R, are shown to be asymptotically exact as R tends to zero or infinity.
This provides a proof of the asymptotic behavior of the boundary height
which previously has been derived by a formal perturbation method. An
asymptotic characterization of the solution in a neighborhood of the boundary is
also given. For a general domain estimates are obtained from a maximum
principle due to Finn in which the symmetric solutions serve as comparison
functions.
