Abstract
Consider a surface x, bounded by a smooth surface in Euclidean n-space and by a smooth curve having its end points on the boundary surface. Let x minimize Dirichlet's integral in the class of all such surfaces x. Then it is known that the trace of x on the boundary surface is a continuous curve. We treat the case where x is a stationary point of Dirichlet's integral and prove continuity in this case.
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Dziuk, G. Über die Stetigkeit teilweise freier Minimalflächen. Manuscripta Math 36, 241–251 (1981). https://doi.org/10.1007/BF01170136
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DOI: https://doi.org/10.1007/BF01170136