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I was wondering, if there is a generalization of the coarea formula to higher order derivatives, which would allow one, for example, to relate the norm of the Hessian of a real-valued function $u$ to an integral over level sets.

Application of the "vector-valued" coarea formula (3rd equation in the Wikipedia entry) to $\nabla u$ is not possible, because it only holds for functions from $\mathbb{R}^n$ to $\mathbb{R}^k$ with $n>k$, which is violated by $\nabla u$.

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Have you found an answer to this question in the meantime? – tomglabst Sep 2 at 12:10
@tomglabst: No. There is, however, the following formula which works for the $L^2$-norm of the Hessian, but doesn't deserve the name coarea formula: $$\int_\Omega |\nabla^2 u|^2\,dx = \sum_{i=1}^n \int_\Omega g_i |\nabla u_{x_i}|\,dx= \ldots $$ where $g_i = |\nabla u_{x_i}|$ and the dots stand for application of the standard coarea formula. – begeistzwerst Sep 2 at 16:40

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