Rademacher's theorem

Template:Short description In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If Template:Mvar is an open subset of Template:Math and Template:Math is Lipschitz continuous, then Template:Mvar is differentiable almost everywhere in Template:Mvar; that is, the points in Template:Mvar at which Template:Mvar 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.Template:Sfnm 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 Charles Morrey.Template:Sfnm In the following, let Template:Mvar denote a Lipschitz-continuous function on Template:Math. The first step of the proof is to show that, for any fixed unit vector Template:Mvar, the Template:Mvar-directional derivative of Template:Mvar exists almost everywhere. This is a consequence of a special case of the Fubini theorem: a measurable set in Template:Math has Lebesgue measure zero if its restriction to every line parallel to Template:Mvar has (one-dimensional) Lebesgue measure zero. Considering in particular the set in Template:Math where the Template:Mvar-directional derivative of Template:Mvar fails to exist (which must be proved to be measurable), the latter condition is met due to the one-dimensional case of Rademacher's theorem.

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

$\underset{{𝐑}^n}{\int }\frac{u(x+h\nu )-u(x)}{h}\zeta (z)d{ℒ}^n(x)=-\underset{{𝐑}^n}{\int }\frac{\zeta (x)-\zeta (x-h\nu )}{h}u(x)d{ℒ}^n(x).$

Using the Lipschitz assumption on Template:Mvar, the dominated convergence theorem can be applied to replace the two difference quotients in the above expression by the corresponding Template:Mvar-directional derivatives. Then, based upon the known linear dependence of the Template:Mvar-directional derivative of Template:Math upon Template:Mvar, the same can be proved of Template:Mvar via the fundamental lemma of calculus of variations.

At this point in the proof, the gradient (defined as the Template:Mvar-tuple of partial derivatives) is guaranteed to exist almost everywhere; for each Template:Mvar, the dot product with Template:Mvar equals the Template:Mvar-directional derivative almost everywhere (although perhaps on a smaller set). Hence, for any countable collection of unit vectors Template:Math, there is a single set Template:Mvar of measure zero such that the gradient and each Template:Math-directional derivative exist everywhere on the complement of Template:Mvar, and are linked by the dot product. By selecting Template:Math 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 Template:Mvar, 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 generalized derivatives.Template:Sfnm Another proof, also via a reduction to the one-dimensional case, uses the technology of approximate limits.Template:Sfnm

Applications

Rademacher's theorem can be used to prove that, for any Template:Math, the Sobolev space Template:Math is preserved under a bi-Lipschitz transformation of the domain, with the chain rule holding in its standard form.Template:Sfnm With appropriate modification, this also extends to the more general Sobolev spaces Template:Math.Template:Sfnm

Rademacher's theorem is also significant in the study of geometric measure theory and rectifiable sets, as it allows the analysis of first-order differential geometry, specifically tangent planes and normal vectors.Template:Sfnm Higher-order concepts such as curvature remain more subtle, since their usual definitions require more differentiability than is achieved by the Rademacher theorem. In the presence of convexity, second-order differentiability is achieved by the Alexandrov theorem, the proof of which can be modeled on that of the Rademacher theorem. In some special cases, the Rademacher theorem is even used as part of the proof.Template:Sfnm

Generalizations

Alberto Calderón proved the more general fact that if Template:Math is an open bounded set in Template:Math then every function in the Sobolev space Template:Math is differentiable almost everywhere, provided that Template:Math.Template:Sfnm Calderón's theorem is a relatively direct corollary of the Lebesgue differentiation theorem and Sobolev embedding theorem. Rademacher's theorem is a special case, due to the fact that any Lipschitz function on Template:Math is an element of the space Template:Math.Template:Sfnm

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

Template:Reflist Sources Template:Refbegin

• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite journal
• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite book

External links

• Template:Cite web (Rademacher's theorem with a proof is on page 18 and further.)