A perturbation result concerning a second solution to the Dirichlet problem for the equation of prescribed mean curvature.

Abstract

A perturbation result concerning a second solution to the Dirichlet problem for the equation of prescribed mean curvature By Norbert Jakobowsky at Aachen 0. Introduction Let B = {(u, v) e IR 2 : u 2 + v 2 < 1}. We look for mappings : B -> R 3 solving the Dirichlet problem (DP) for the equation of prescribed mean curvaturewhere H is a given real bounded function in IR 3 , the exterior product in R 3 , and z 0 denotes the boundary data. Solutions to (DP) can be obtained äs critical points of the functional3 )nL°° and divß(x) = 3H(x). As a suitable choice for Q we set (0.3) Q(x) = ß H (x) = (7 H (s, x 2 , xjds, ] H(x i9 s, x 3 )ds, 7 Ä(^, x 2 , s)ds] , which reduces to ß(x) = /ix in case of H = const, and define E H = E Q . Generalizing results of Heinz [5], and Werner [27], for H = const, Hildebrandt [8], [9] found that under certain conditions on z 0 and H the functional E H admits some relative minimizer: Theorem 1. (i) Suppose z 0 e W^2(B 9 R 3 )nL°°, He (0.4) H^olioo;B<l 5 Brought to you by | University of Iowa Libraries Authenticated Download Date | 6/1/15 6:29 AM