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{{MSC|28A}}
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{{MSC|49Q15|49Q20,49Q05,28A75,32C30,58A25,58C35}}
  
 
[[Category:Classical measure theory]]
 
[[Category:Classical measure theory]]
  
 
{{TEX|done}}
 
{{TEX|done}}
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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|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.
  
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==Definition==
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=History=
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An outer measure is a [[Set function|set function]] $\mu$ such that
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=Measure theoretic concepts=
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* Its domain of definition is an hereditary [[Ring of sets|$\sigma$-ring]] (also called $\sigma$-ideal) of subsets of a given space $X$, i.e. a $\sigma$-ring $\mathcal{R}\subset \mathcal{P} (X)$ with the property that for every $E\in \mathcal{R}$ all subsets of $E$ belong to $\mathcal{R}$;
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==Caratheodory construction==
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* Its range is the extended real half-line $[0, \infty$];
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===Hausdorff measures===
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* $\mu (\emptyset) =0$ and $\mu$ is ''$\sigma$-subadditive'' (also called ''countably subadditive''), i.e. for every countable family $\{E_i\}\subset \mathcal{R}$ the following inequality holds:
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===Fractals===
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\[
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=Rectifiable sets=
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\mu \left(\bigcup_i E_i\right) \leq \sum_i \mu (E_i)\, .
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==Besicovitch's works==
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\]
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===One-dimensional sets===
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Cp. with Section 10 of {{Cite|Ha}} and with Section 1.1 of {{Cite|EG}}.
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==General dimension and codimension==
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The most common outer measures are defined on the full space $\mathcal{P} (X)$ of subsets of $X$. Indeed observe that, if an hereditary $\sigma$-ring is also an [[Algebra of sets|algebra]], then it must contain $X$ and hence it coincides necessarily with $\mathcal{P} (X)$.
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===Besicovitch-Federer projection theorem===
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===Marstrand's theorem===
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===Measurable sets===
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===Besicovitch-Preiss theorem===
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There is a commonly used procedure to derive a measure from an outer measure $\mu$ on an hereditary $\sigma$-ring $\mathcal{R}$ (cp. with Section 11 of {{Cite|Ha}} and Section 1.1 of {{Cite|EG}}).
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====Tangent measures====
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=Caccioppoli sets=
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'''Definition 1'''
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==Functions of bounded variation==
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If $\mu:\mathcal{R}\to [0, \infty]$ is an outer measure, then a set $M\in \mathcal{R}$ is called $\mu$-measurable if
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==Plateau's problem in codimension 1==
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\[
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===Existence===
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\mu (A\cap M) + \mu (A\setminus M) = \mu (A) \qquad \forall A\in \mathcal{R}\, .
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==Regularity theory==
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\]
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===Bernstein's problem===
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====Simons' cone====
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'''Theorem 2'''
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===De Giorgi's $\varepsilon$-regularity theorem===
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If $\mu:\mathcal{R}\to [0, \infty]$ is an outer measure, then the class $\mathcal{M}$ of $\mu$-measurable sets is a $\sigma$-ring and $\mu$ is countably additive on $\mathcal{M}$, i.e.
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===Simons' inequality and solution of the Bernstein's problem===
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\[
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===Federer's estimate of the singular set===
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\mu \left(\bigcup_i E_i\right) = \sum_i \mu (E_i)
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===Simon's rectifiability theorem===
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\]
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==Mumford Shah conjecture==
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whenever $\{E_i\}\subset \mathcal{M}$ is a countable collection of pairwise disjoint sets.
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=Currents=
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==Federer-Fleming theory==
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When $\mathcal{R} = \mathcal{P} (X)$, then it follows trivially from the definition that $X\in \mathcal{M}$: thus $\mathcal{M}$ is in fact a [[Algebra of sets|$\sigma$-algebra]]. Therefore $(X, \mathcal{M}, \mu)$ is a [[Measure space|measure space]].
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===Compactness for integral currents===
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===Deformation theorem===
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==Caratheodory criterion==
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==Plateau's problem in any codimension==
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An important class of outer measures on metric spaces $X$ are the ones satisfying the so-called Caratheodory criterion (called ''metric outer measures'' or [[Caratheodory measure|Caratheodory measures]]): for such $\mu$ the [[Borel set|Borel sets]] are $\mu$-measurable.
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==Regularity theory==
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===Almgren's $\varepsilon$-regularity theorem===
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'''Definition 3'''
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===Almgren's big regularity paper===
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An outer measure $\mu: \mathcal{P} (X) \to [0, \infty]$ on a metric space $(X,d)$ is a [[Caratheodory measure]] if
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==Currents in metric spaces==
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\[
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=Varifolds=
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\mu (A\cup B) = \mu (A) + \mu (B)
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==General theory==
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\]
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==Rectifiable and integral varifolds==
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for every pair of sets $A, B\subset X$ which have positive distance (i.e. such that $\inf \{d(x,y): x\in A, y\in B\} > 0$).
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==Regularity theory==
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===Allard's rectifiability theorem===
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'''Theorem 4'''
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===Allard's $\varepsilon$-regularity theorem===
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If $\mu$ is a Caratheodory measure, then every Borel set is $\mu$-measurable.
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=Calculus of variations in the large=
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==Pitts' theory==
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Cp. with Theorem 5 of {{Cite|EG}}.
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===Schoen-Simon curvature estimates===
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==Smith's theorem and generalizations==
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===Regular and Borel regular outer measures===
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===Applications to topology===
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Several authors call regular those outer measures $\mu$ on $\mathcal{P} (X)$ such that for every $E\subset X$ there is a $\mu$-measurable set $F$ with $E\subset F$ and $\mu (E) = \mu (F)$. They moreover call $\mu$ Borel regular if the Borel sets are $\mu$-measurable and  for every $E\subset X$ there is a ''Borel set'' $G$ with $E\subset G$ and $\mu (E) = \mu (G)$. Cp. with Section 1.1 of {{Cite|EG}}.
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===The Willmore conjecture===
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=Uniqueness of tangent cones=
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==Constructions of outer measures==
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==White's theorem==
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===Outer measures induced by measures===
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==Simon's theorem==
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A common procedure to construct outer measures $\mu$ from a set function $\nu$ is the following.
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===Lojasievicz inequality===
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=Soap films=
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'''Definition 5'''
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==Almgren's $\varepsilon-\delta$ minimal sets==
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If $\mathcal{C}$ is class of subsets of $X$ containing the empty set and $\nu : \mathcal{C}\to [0, \infty]$ a set function with $\nu (\emptyset) =0$, for every set $A\subset X$ we define
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==Taylor's theorem==
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\begin{equation}\label{e:extension}
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==Double-bubble conjecture==
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\mu (A) = \inf \left\{ \sum_i \mu (E_i) : \{E_i\}_{i\in \mathbb N} \subset \mathcal{C} \mbox{ covers $A$}\right\}\, .
 
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\end{equation}
 
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Here we use the convention that $\mu (A) = \infty$ when $A\not \in \mathcal{H}$ (cp. for instance with {{Cite|Mu}}).
 
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Some authors define such $\mu$ on the hereditary $\sigma$-ring $\mathcal{H}$ of subsets of $X$ for which there is a countable covering in $\mathcal{C}$ (cp. with Section 10 of {{Cite|Ha}}).
 
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'''Theorem 6'''
 
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If $\nu$ and $\mu$ are as in Definition 5, then $\mu$ is an outer measure on $\mathcal{P} (X)$. If in addition
 
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* $\mathcal{C}$ is a ring and $\nu$ is a finitely additive set function, then $\mu (E) = \nu (E)$ for every $E\in \mathcal{C}$;
 
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* $\mathcal{C}$ is a $\sigma$-ring and $\nu$ is countably additive, then the elements of $\mathcal{C}$ are $\mu$-measurable.
 
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Cp. with Theorem A of Section 10 and Theorem A in Section 12 of {{Cite|Ha}} (NB: the proof given in {{Cite|Ha}} of $\sigma$-subadditivity of $\mu$ does not use the assumption that $\nu$ is finitely additive).
 
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===Caratheodory constructions of metric outer measures===
 
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A second common procedure yields metric outer measures in metric spaces $(X, d)$ and goes as follows.
 
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'''Definition 7'''
 
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If $\mathcal{C}$ is class of subsets of $X$ containing the empty set, $\nu : \mathcal{C}\to [0, \infty]$ a set function with $\nu (\emptyset) =0$ and $\delta\in ]0, \infty]$, then for every $A\subset X$ we define
 
−
\[
 
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\mu^\delta (A) := \inf \left\{ \sum_i \mu (E_i) : \{E_i\}_{i\in \mathbb N} \subset \mathcal{C} \mbox{ covers $A$ and ${\rm diam}\, (E_i) \leq \delta$}\right\}\,
 
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\]
 
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and
 
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\[
 
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\mu (A) := \lim_{\delta\downarrow 0}\; \mu^\delta (A)\, .
 
−
\]
 
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Observe that the latter limit exists because $\mu^\delta (A)$ is a nonincreasing function of $\delta$.
 
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This construction is often called ''Caratheodory construction''. See Section 2.1 of {{Cite|KP}} (cp. also with {{Cite|Fe}}).
 
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'''Theorem 8'''
 
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Let $\nu$ and $\mu$ be as in Definition 7. Then $\mu$ is a metric outer measure.
 
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(Cp. with Claims 1,2 and 3 in the proof of Theorem 1 in Section 2.1 of {{Cite|EG}}: although the reference handles the cases of Hausdorff measures, the proof extends verbatim to the setting above).
 
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'''Remark 9'''
 
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The [[Hausdorff measure]] $\mathcal{H}^\alpha$ is given by such $\mu$ as in Definition 7 when we choose $\mathcal{C} = \mathcal{P} (X)$ and $\nu (A) = c_\alpha ({\rm diam}\, (A))^\alpha$ (where $c_\alpha$ is an appropriate normalization constant).
 
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==Examples==
 
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Very common examples of outer measures are
 
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* The Lebesgue outer measure on $\mathbb R^n$, see [[Lebesgue measure]];
 
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* The Haudorff $\alpha$-dimensional measures on a metric space $(X,d)$, see [[Hausdorff measure]];
 
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* The spherical $\alpha$-dimensional measures on a metric space $(X,d)$, see Section 2.1.2 of {{Cite|KP}};
 
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* The Gross measures, the Caratheodory measures, the integral-geometric measures and the Gillespie measures in $\mathbb R^n$, see Sections 2.1.3-2.1.4-2.1.5 of {{Cite|KP}} (cp. also with {{Cite|Fe}}).
 
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==References==
 
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{|
 
−
|-
 
−
|valign="top"|{{Ref|EG}}||  L.C. Evans, R.F. Gariepy, "Measure theory  and fine properties of  functions" Studies in Advanced Mathematics. CRC  Press, Boca Raton, FL,  1992. {{MR|1158660}} {{ZBL|0804.2800}}
 
−
|-
 
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|valign="top"|{{Ref|Fe}}||  H. Federer, "Geometric measure  theory". Volume 153 of Die Grundlehren  der mathematischen  Wissenschaften. Springer-Verlag New York Inc., New  York, 1969.  {{MR|0257325}} {{ZBL|0874.49001}}
 
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|-
 
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|valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
 
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|-
 
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|valign="top"|{{Ref|KP}}|| S. G. Krantz, H. Parks, "Geometric Integration Theory", Birkhäuser (2008).
 
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|-
 
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|valign="top"|{{Ref|Ma}}||    P. Mattila, "Geometry of sets  and measures in euclidean spaces".    Cambridge Studies in Advanced  Mathematics, 44. Cambridge University    Press, Cambridge,  1995.  {{MR|1333890}} {{ZBL|0911.28005}}
 
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|-
 
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|}
 

Revision as of 20:20, 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

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Camillo.delellis/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Camillo.delellis/sandbox&oldid=28065
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