Vol. 88, No. 2, 1980

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Height estimates for exterior problems of capillarity type

Bruce Edward Turkington

Vol. 88 (1980), No. 2, 517–540

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.

Mathematical Subject Classification 2000
Primary: 35J65
Secondary: 49F10
Received: 28 March 1979
Published: 1 June 1980
Bruce Edward Turkington