A tour of Lipschitz truncations

Bianca Stroffolini (Napoli, Italy)

The purpose of the Lipschitz truncation is to regularize a given function by a Lipschitz continuous one by changing it only on a small  bad set. It is crucial for the applications that the function is not changed globally, which rules out the possibility of convolutions.

The Lipschitz truncation technique was introduced by Acerbi-Fusco to show lower semicontinuity of certain variational integrals.

Since then this technique has been successfully applied in many different areas: biting lemmas, existence theory  and regularity results of non-linear elliptic PDE . It was also successfully applied in the framework of non-Newtonian fluids of power law type and even in the context of numerical analysis.

I will try to present some Lipschitz truncations Lemmas.
As an application, existence/ regularity of solutions of PDEs will be discussed.

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