Non-parametric convex hypersurfaces with a curvature restriction
Rolf Schneider (1983)
Annales Polonici Mathematici
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Rolf Schneider (1983)
Annales Polonici Mathematici
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Peter Kohlmann (1995)
Journal für die reine und angewandte Mathematik
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Quiyi Dai, Neil S. Trudinger, Xu-Jia Wang (2012)
Journal of the European Mathematical Society
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We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a sharp Harnack inequality for the minimal surface equation, which is crucial for our proof of the weak continuity. As an application we prove the existence of weak solutions...
Jan Rataj (2004)
Czechoslovak Mathematical Journal
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The information contained in the measure of all shifts of two or three given points contained in an observed compact subset of is studied. In particular, the connection of the first order directional derivatives of the described characteristic with the oriented and the unoriented normal measure of a set representable as a finite union of sets with positive reach is established. For smooth convex bodies with positive curvatures, the second and the third order directional derivatives...
Guy Bouchitté, Giuseppe Buttazzo, Ilaria Fragalà (1997)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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Joo, I., Stacho, L.L. (1981)
Publications de l'Institut Mathématique. Nouvelle Série
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Eduardo H. A. Gonzales, Umberto Massari, Italo Tamanini (1993)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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The existence of a singular curve in is proven, whose curvature can be extended to an 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.
Xu-Jia Wang (2014)
Journal of the European Mathematical Society
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The convexity of level sets of solutions to the mean curvature equation is a long standing open problem. In this paper we give a counterexample to it.
Lohkamp, Joachim (1998)
Documenta Mathematica
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