Calculus of Variations and Geometric Measure Theory
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G. Mingione

Nonlinear aspects of Calderón-Zygmund theory

created by mingione on 29 Sep 2012


Published Paper

Inserted: 29 sep 2012
Last Updated: 29 sep 2012

Journal: J. Deutsch. Math. Ver.
Volume: 112
Pages: 159-191
Year: 2010


Calderón-Zygmund theory is classically a linear fact and amounts to get sharp properties on integrability and differentiability of solutions of linear equations starting from that of the given data. A typical question is for instance: Given the Poisson equation $ -\triangle u=\mu, $ in which Lebesgue space do $Du$ or $D^2u$ lie if we assume that $\mu \in L^\gamma$ for some $\gamma\geq 1$? Questions of this type have been traditionally answered using the theory of singular integrals and using Harmonic Analysis methods, which perfectly fit the case of linear equations. The related results lie at the core of nowadays analysis of pdes as they often provide the first regularity information after which further qualitative properties of solutions can be established. In the last years there has anyway been an ever growing number of results concerning nonlinear equations: put together, they start shaping what we may call a nonlinear Calderón-Zygmund theory. This means a theory which reproduces for non-linear equations the results and phenomena known for linear ones, without necessarily appealing to linear techniques and tools. The approaches are in this case suited to the special equations under consideration. Yet, although bypassing general Harmonic Analysis tools, in some way they preserve the general spirit of some the basic Harmonic Analysis ideas, applying them directly at a pde level. This is a report on some of the main results available in this context.

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