We provide estimates for the gradient growth of surfaces with prescribed mean curvature in 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.
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Frank Müller, Growth Estimates for the Gradient of an <em>H</em>-Surface Near Singular Points of the Boundary Configuration. Z. Anal. Anwend. 28 (2009), no. 1, pp. 87–102DOI 10.4171/ZAA/1374