Comparison between the generalized mean curvature according to Allard and Federer's mean curvature measure

E. Ossanna

Displaying similar documents to “Comparison between the generalized mean curvature according to Allard and Federer's mean curvature measure”

Non-parametric convex hypersurfaces with a curvature restriction

Rolf Schneider (1983)

Annales Polonici Mathematici

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Compact convex bodies with one curvature measure near the surface measure.

Peter Kohlmann (1995)

Journal für die reine und angewandte Mathematik

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The mean curvature measure

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...

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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 $ℝ^{d}$ 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...

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Guy Bouchitté, Giuseppe Buttazzo, Ilaria Fragalà (1997)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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An upper estimation for the eigenfrequencies of vibrating Liapunoff bodies (first boundary value problem).

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 $R^{2}$ 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.

Counterexample to the convexity of level sets of solutions to the mean curvature equation

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.

Curvature contents of geometric spaces.

Lohkamp, Joachim (1998)

Documenta Mathematica

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Curvature bounds for neighborhoods of self-similar sets

Steffen Winter (2011)

Commentationes Mathematicae Universitatis Carolinae

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In some recent work, fractal curvatures $C_{k}^{f} (F)$ and fractal curvature measures $C_{k}^{f} (F, \cdot)$ , $k = 0, ..., d$ , have been determined for all self-similar sets $F$ in $ℝ^{d}$ , for which the parallel neighborhoods satisfy a certain regularity condition and a certain rather technical curvature bound. The regularity condition is conjectured to be always satisfied, while the curvature bound has recently been shown to fail in some concrete examples. As a step towards a better understanding of its meaning, we discuss several equivalent...