Rademacher's theorem

In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If U is an open subset of Rⁿ and f: U → R^m is Lipschitz continuous, then f is differentiable almost everywhere in U; that is, the points in U at which f is not differentiable form a set of Lebesgue measure zero. Differentiability here refers to infinitesimal approximability by a linear map, which in particular asserts the existence of the coordinate-wise partial derivatives.

Sketch of proof

The one-dimensional case of Rademacher's theorem is a standard result in introductory texts on measure-theoretic analysis.^[1] In this context, it is natural to prove the more general statement that any single-variable function of bounded variation is differentiable almost everywhere. (This one-dimensional generalization of Rademacher's theorem fails to extend to higher dimensions.)

One of the standard proofs of the general Rademacher theorem was found by

The second step of Morrey's proof establishes the linear dependence of the v-directional derivative of u upon v. This is based upon the following identity:

${\displaystyle \int _{\mathbf {R} ^{n}}{\frac {u(x+h\nu )-u(x)}{h}}\zeta (z)\,d{\mathcal {L}}^{n}(x)=-\int _{\mathbf {R} ^{n}}{\frac {\zeta (x)-\zeta (x-h\nu )}{h}}u(x)\,d{\mathcal {L}}^{n}(x).}$

Using the Lipschitz assumption on u, the

At this point in the proof, the gradient (defined as the n-tuple of partial derivatives) is guaranteed to exist almost everywhere; for each v, the dot product with v equals the v-directional derivative almost everywhere (although perhaps on a smaller set). Hence, for any countable collection of unit vectors v₁, v₂, ..., there is a single set E of measure zero such that the gradient and each v_i-directional derivative exist everywhere on the complement of E, and are linked by the dot product. By selecting v₁, v₂, ... to be dense in the unit sphere, it is possible to use the Lipschitz condition to prove the existence of every directional derivative everywhere on the complement of E, together with its representation as the dot product of the gradient with the direction.

Morrey's proof can also be put into the context of

Rademacher's theorem can be used to prove that, for any p ≥ 1, the Sobolev space W^1,p(Ω) is preserved under a bi-Lipschitz transformation of the domain, with the chain rule holding in its standard form.^[5] With appropriate modification, this also extends to the more general Sobolev spaces W^k,p(Ω).^[6]

Rademacher's theorem is also significant in the study of

There is a version of Rademacher's theorem that holds for Lipschitz functions from a Euclidean space into an arbitrary metric space in terms of metric differentials instead of the usual derivative.

See also

• Pansu derivative

References

Sources

• Heinonen, Juha (2004). "Lectures on Lipschitz Analysis" (PDF). Lectures at the 14th Jyväskylä Summer School in August 2004. (Rademacher's theorem with a proof is on page 18 and further.)