Geometric Measure Theory

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Geometric Measure Theory studies the geometric properties of the measure of sets and generalizes concepts of differential geometry to surfaces with singularities.

Geometric Measure Theory

The birth of the subject is usually set by the pioneering works of Besicovitch on the 1-dimensional sets with finite 1-dimensional Hausdorff measure. However the subject boomed in the fifties and the sixties thanks to the genius of mathematicians like De Giorgi, Federer, Fleming, Reifenberg and Almgren, who used methods from measure theory to settle long standing questions in the calculus of variations.

Plateau's problem A central question in geometric measure theory is that of minimizing the area among all surfaces with a given contour.

W Jones , Thomas H. Geometry of measures in Rn: distribution David Preiss.

T. Toro - Geometry of measures and applications (Part 1)

Caffarelli , Avner Friedman. Boundary behavior of harmonic functions in non-tangentially accessible domains David S. Jerison , Carlos E. Mutual absolute continuity of interior and exterior harmonic measure implies rectifiability Jonas Azzam , Mihalis Mourgoglou , Xavier Tolsa.


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On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1. We develop an analysis framework that extends this result to currents in general. In the second problem, we study a notion of average shape defined as the median of currents.

5th Edition

In the corresponding discrete version, the median is computed efficiently using LP. Skip to main content. Toggle menu Go to search page. Search Field.


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