Difference between revisions of "Rademacher theorem"
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| − | A theorem proved by H. Rademacher about the differentiability of [[Lipschitz | + | A theorem proved by H. Rademacher about the differentiability of [[Lipschitz Function|Lipschitz functions]] . |
'''Theorem''' | '''Theorem''' | ||
Let $U$ be an open subset of $\mathbb R^n$ and $f:U\to \mathbb R^k$ a Lipschitz function, i.e. such that there is a constant $C$ with | Let $U$ be an open subset of $\mathbb R^n$ and $f:U\to \mathbb R^k$ a Lipschitz function, i.e. such that there is a constant $C$ with | ||
\[ | \[ | ||
| − | |f(x)-f(y)|\leq C|x-y|\qquad \mbox{for every | + | |f(x)-f(y)|\leq C|x-y|\qquad \mbox{for every } x,y\in U\, . |
\] | \] | ||
Then $f$ is differentiable almost everywhere (with respect to the Lebesgue measure $\lambda$). That is, there is a set $E\subset U$ with $\lambda (U\setminus E) = 0$ and such that for every $x\in E$ there is a linear function $L_x:\mathbb R^n\to \mathbb R^k$ with | Then $f$ is differentiable almost everywhere (with respect to the Lebesgue measure $\lambda$). That is, there is a set $E\subset U$ with $\lambda (U\setminus E) = 0$ and such that for every $x\in E$ there is a linear function $L_x:\mathbb R^n\to \mathbb R^k$ with | ||
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For a proof see Theorem 2.14 of {{Cite|AFP}} or {{Cite|EG}}. | For a proof see Theorem 2.14 of {{Cite|AFP}} or {{Cite|EG}}. | ||
| − | The same conclusion holds also for maps in the Sobolev class $W^{1,p}$ if $p$ is strictly larger then the dimension of the domain. | + | The same conclusion holds also for maps in the Sobolev class $W^{1,p}$ if $p$ is strictly larger then the dimension of the domain. A closely related, and more general, result is [[Stepanov theorem]] and |
| + | indeed some author use the terminology "Rademacher-Stepanov theorem". | ||
| − | === | + | ===References=== |
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Latest revision as of 14:09, 2 May 2014
2020 Mathematics Subject Classification: Primary: 26B05 Secondary: 26B35 [MSN][ZBL]
A theorem proved by H. Rademacher about the differentiability of Lipschitz functions .
Theorem Let $U$ be an open subset of $\mathbb R^n$ and $f:U\to \mathbb R^k$ a Lipschitz function, i.e. such that there is a constant $C$ with \[ |f(x)-f(y)|\leq C|x-y|\qquad \mbox{for every } x,y\in U\, . \] Then $f$ is differentiable almost everywhere (with respect to the Lebesgue measure $\lambda$). That is, there is a set $E\subset U$ with $\lambda (U\setminus E) = 0$ and such that for every $x\in E$ there is a linear function $L_x:\mathbb R^n\to \mathbb R^k$ with \[ \lim_{y\to x} \frac{f (x) - f(y) - L_x (y-x)}{|y-x|} \; =\; 0\, . \]
For a proof see Theorem 2.14 of [AFP] or [EG].
The same conclusion holds also for maps in the Sobolev class $W^{1,p}$ if $p$ is strictly larger then the dimension of the domain. A closely related, and more general, result is Stepanov theorem and indeed some author use the terminology "Rademacher-Stepanov theorem".
References
| [AFP] | L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001 |
| [EG] | L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800 |
Rademacher theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rademacher_theorem&oldid=27663