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Sobolev regularity and an enhanced Jensen inequality

Mark A. Peletier·Robert Planqué·Matthias Röger
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Abstract

We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1,2}(\R^n)$. This criterion consists of comparing the value of a functional $\int f(u)$ with the values of the same functional applied to convolutions of $u$ with a Dirac sequence. The difference of these values converges to zero as the convolutions approach $u$, and we prove that the rate of convergence to zero is connected to regularity: $u\in W^{1,2}$ if and only if the convergence is sufficiently fast. We finally apply our criterium to a minimization problem with constraints, where regularity of minimizers cannot be deduced from the Euler-Lagrange equation.

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Mark A. Peletier, Robert Planqué, Matthias Röger. 2007-01-15. Sobolev regularity and an enhanced Jensen inequality. https://arxiv.org/abs/math/0701412

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