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The equality of mixed partial derivatives under weak differentiability conditions. (English) Zbl 1393.26010

In this paper, the author reviews and develops some known results on the equality of mixed partial derivatives, specially those due to Mikusiński dealing with equality at a given point and to Tolstov concerning equality almost everywhere.
Results about equality of mixed partial derivatives have a long history and the first correct proofs are due to Cauchy and to H. A. Schwarz. A typical result on this topic is the one due to Peano that reads as follows:
\(\bullet\) Let \(O=(a,b)\times (c,d)\subset \mathbb{R}^{2}\) and \(f: O\to\mathbb{R}\). Suppose that \(\partial_{1}f\), \(\partial_{2}f\) and \(\partial_{2}\partial_{1}f\) exist on \(O\) and that the latter is continuous at \((x_{0},y_{0})\). Then \(\partial_{1}\partial_{2}f(x_{0},y_{0})=\partial_{2}\partial_{1}f(x_{0},y_{0})\).
A result that improves Peano’s can be obtained by means of the concept of strong differentiation, introduced by himself:
A function \(f: B\to \mathbb{R}^{c}\), \(B\subset \mathbb{R}^{a}\times \mathbb{R}^{b}\), \((x,z)\to f(x,z)\) is said to be partially strongly differentiable with respect to \(x\) at \((x_{0},z_{0})\in\bar{B}\) with differential \(\partial_{1}f(x_{0},z_{0})\) if for every \(\varepsilon>0\), there is a \(\delta>0\) such that for every \(x_{1}\), \(x_{2}\), \(z\) with \(\|x_{1}-x_{0}\|<\delta\), \(\|x_{2}-x_{0}\|<\delta\), \(\|z-z_{0}\|<\delta\), \((x_{1},z)\in B\), \((x_{2},z)\in B\), \[ \|f(x_{2},z)-f(x_{1},z)-\partial_{1}f(x_{0},z_{0})(x_{2}-x_{1})\|\leq \varepsilon \|x_{2}-x_{1}\|. \]

Peano’s theorem demands the existence of \(\partial_{2}\partial_{1}f\) in a neighborhood of \((x_{0},y_{0})\) and its continuity there. So \(\partial_{1}f\) is partially strongly differentiable with respect to \(y\) at \((x_{0},y_{0})\). This is the motivation of the following result due to Mikusiński.
\(\bullet\) Let \(f: O\to\mathbb{R}\). Suppose that the partial derivative \(\partial_{f}f\) exists on \(O\) and that it is partially strongly differentiable with respect to \(y\) at \((x_{0},y_{0})\). Then, denoting by \(A\subset O\) the subset where \(\partial_{2}f\) exists, provided \((x_{0},y_{0})\in\bar A\), \(\partial_{2}f(:=\partial_{2}f|_{A})\) is partially strongly differentiable with respect to \(x\) at \((x_{0},y_{0})\) and \(\partial_{1}\partial_{2}f(x_{0},y_{0})=\partial_{2}\partial_{1}f(x_{0},y_{0})\).
One may also ask to what extend the equality of mixed partial derivatives holds for functions which admit second derivatives almost everywhere. The Russian mathematician G. P. Tolstov clarified several questions about this problem. Most of his results concern counterexamples. For instance, he showed:
\(\bullet\) There exists a function \(f\in C^{1}(O,\mathbb{R})\), the mixed second derivatives of which exist at every point of \(O\), but such that \(\partial_{2}\partial_{1}f\neq \partial_{1}\partial_{2}f\) on a set \(P\subset O\) of positive measure.
\(\bullet\) There exists a function \(f\in C^{1}(O,\mathbb{R})\), the mixed second derivatives of which exist almost everywhere in \(O\), and such that \(\partial_{2}\partial_{1}f\neq \partial_{1}\partial_{2}f\) almost everywhere in \(O\).
As for positive results, he proved, e.g., the following theorem:
\(\bullet\) Let \(f: [a,b]\times [c,d]\to\mathbb{R}\) such that \(f(x,\cdot): [c,d]\to\mathbb{R}\) and \(f(\cdot,y): [a,b]\to\mathbb{R}\) are absolutely continuous for every \(x\in[a,b]\) and \(y\in [c,d]\), respectively. The following properties are equivalent:
(i) There is \(e_{1}\subset (a,b)\), \(|e_{1}|=b-a\), such that for \(x\in e_{1}\), \(\partial_{1}f(x,\cdot)\) exists for every \(y\). Moreover, it is absolutely continuous over \([c,d]\), and \(\partial_{2}\partial_{1}f\in L^{1}([a,b]\times [c,d])\).
(ii) There is \(e_{2}\subset (c,d)\), \(|e_{2}|=d-c\), such that for \(y\in e_{2}\), \(\partial_{2}f(\cdot,y)\) exists for every \(x\). Moreover, it is absolutely continuous over \([a,b]\), and \(\partial_{1}\partial_{2}f\in L^{1}([a,b]\times [c,d])\).
Suppose they hold true. Then, there is a subset \(E\subset e_{1}\times e_{2}\), \(|E|=(b-a)(d-c)\), such that on \(E\) the function \(f\) is differentiable, \(\partial_{2}\partial_{1} f(x,y)\), \(\partial_{1}\partial_{2}f(x,y)\) exist, and \[ \partial_{2}\partial_{1}f=\partial_{1}\partial_{2}f. \]
Another type of results can be obtained if one introduces some Lipschitz condition on \(f\) or on the partial derivatives \(\partial_{1}f\), \(\partial_{2}f\). A function \(f: U\to\mathbb{R}\), \(U\) an open set in \(\mathbb{R}^{2}\), is differentiable with Lipschitz differential, or \(f\in C^{1,1}\), if \(df: U\to\mathbb{R}^{2}\) is Lipschitz. Then, one has:
\(\bullet\) If \(f\in C^{1,1}_{\mathrm{loc}}(U)\), then there is a set \(E\subset U\) of full measure, such that \(\partial_{2}\partial_{1}f\) and \(\partial_{1}\partial_{2}f\) exist on \(E\), \(f\) is differentiable and \(\partial_{2}\partial_{1}f\) on \(E\).
Using Rademacher’s theorem one gets:
\(\bullet\) Let \(f: U\to\mathbb{R}\), \(f\in C^{1,1}_{\mathrm{loc}}(U)\). Then, \(f\) is twice differentiable almost everywhere on \(U\) and in such differentiability set \(\partial_{2}\partial_{1}f=\partial_{1}\partial_{2}f\).
It is an interesting fact that the above results with Lipschitz type conditions have applications to differential geometry and to general relativity theory.

MSC:

26B05 Continuity and differentiation questions
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
26A16 Lipschitz (Hölder) classes
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