On the existence of surfaces with prescribed mean curvature and boundary in higher dimensions

Frank Duzaar

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The existence of a singular curve in $R^{2}$ is proven, whose curvature can be extended to an $L^{2}$ function. The curve is the boundary of a two dimensional set, minimizing the length plus the integral over the set of the extension of the curvature. The existence of such a curve was conjectured by E. De Giorgi, during a conference held in Trento in July 1992.