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We provide estimates for the gradient growth of surfaces with prescribed mean curvature in R3 near boundary points, which are mapped onto singular points of the boundary configuration. For corners of a Jordan arc, such estimates were provided by G. Dziuk [Analysis 1 (1981), 63–81]. We consider meeting points of a Jordan arc and a support manifold, as appearing in a partially free boundary problem (see G. Dziuk [Manuscr. Math. 35 (1981), 105–123] for the minimal surface case), and edge-type singularities of a support manifold. In subsequent papers, these results shall be used to derive asymptotic expansions of surfaces with prescribed mean curvature near such singular points.

Growth Estimates for the Gradient of an $H$-Surface Near Singular Points of the Boundary Configuration