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A geometrical approach to monotone functions in \(\mathbb R^n\)

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Abstract.

The paper is concerned with the fine properties of monotone functions on \(\mathbb{R}^n\). We study the continuity and differentiability properties of these functions, the approximability properties, the structure of the distributional derivatives and of the weak Jacobians. Moreover, we exhibit an example of a monotone function u which is the gradient of a \(C^{1,\alpha}\) convex function and whose weak Jacobian Ju is supported on a purely unrectifiable set.

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Received October 9, 1996; in final form April 21, 1997

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Alberti, G., Ambrosio, L. A geometrical approach to monotone functions in \(\mathbb R^n\) . Math Z 230, 259–316 (1999). https://doi.org/10.1007/PL00004691

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  • DOI: https://doi.org/10.1007/PL00004691

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