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Difference between revisions of "User:Camillo.delellis/sandbox"

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==Taylor's theorem==
 
==Taylor's theorem==
 
==Double-bubble conjecture==
 
==Double-bubble conjecture==
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=References=

Revision as of 20:22, 17 October 2012

2020 Mathematics Subject Classification: Primary: 49Q15 Secondary: 49Q2049Q0528A7532C3058A2558C35 [MSN][ZBL]


An area of analysis concerned with solving geometric problems via measure-theoretic techniques. The canonical motivating physical problem is probably that investigated experimentally by J. Plateau in the nineteenth century: Given a boundary wire, how does one find the (minimal) soap film which spans it? Slightly more mathematically: Given a boundary curve, find the surface of minimal area spanning it. (Cf. also Plateau problem.) The many different approaches to solving this problem have found utility in most areas of modern mathematics and geometric measure theory is no exception: techniques and ideas from geometric measure theory have been found useful in the study of partial differential equations, the calculus of variations, geometric analysis, harmonic analysis, and fractals.

History

Measure theoretic concepts

Caratheodory construction

Hausdorff measures

Fractals

Rectifiable sets

Besicovitch's works

One-dimensional sets

General dimension and codimension

Besicovitch-Federer projection theorem

Marstrand's theorem

Besicovitch-Preiss theorem

Tangent measures

Caccioppoli sets

Functions of bounded variation

Plateau's problem in codimension 1

Existence

Regularity theory

Bernstein's problem

Simons' cone

De Giorgi's $\varepsilon$-regularity theorem

Simons' inequality and solution of the Bernstein's problem

Federer's estimate of the singular set

Simon's rectifiability theorem

Mumford Shah conjecture

Currents

Federer-Fleming theory

Compactness for integral currents

Deformation theorem

Plateau's problem in any codimension

Regularity theory

Almgren's $\varepsilon$-regularity theorem

Almgren's big regularity paper

Currents in metric spaces

Varifolds

General theory

Rectifiable and integral varifolds

Regularity theory

Allard's rectifiability theorem

Allard's $\varepsilon$-regularity theorem

Calculus of variations in the large

Pitts' theory

Schoen-Simon curvature estimates

Smith's theorem and generalizations

Applications to topology

The Willmore conjecture

Uniqueness of tangent cones

White's theorem

Simon's theorem

Lojasievicz inequality

Soap films

Almgren's $\varepsilon-\delta$ minimal sets

Taylor's theorem

Double-bubble conjecture

References

How to Cite This Entry:
Camillo.delellis/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Camillo.delellis/sandbox&oldid=28496