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Archive for Rational Mechanics and Analysis

, Volume 227, Issue 2, pp 663–714 | Cite as

Nonlinear Calderón–Zygmund Theory in the Limiting Case

  • Benny Avelin
  • Tuomo Kuusi
  • Giuseppe Mingione
Article

Abstract

We prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems. Specifically, we deal with the limiting case of the classical theory of Calderón and Zygmund in the setting of nonlinear, possibly degenerate equations and we show a complete linearization effect with respect to the differentiability of solutions. A prototype of the results obtained here tells for instance that if
$$-{\rm div} \, (|Du|^{p-2}Du)=\mu \quad \mbox{in} \ \Omega\subset\mathbb{R}^n,$$
with \({\mu}\) being a Borel measure with locally finite mass on the open subset \({\Omega\subset \mathbb{R}^n}\) and \({p > 2-1/n}\), then
$$|Du|^{p-2}Du \in W^{\sigma, 1}_{\rm{loc}}(\Omega)\quad \mbox{for \, every} \ \sigma \in (0,1).$$
The case \({\sigma=1}\) is obviously forbidden already in the classical linear case of the Poisson equation \({-\triangle u=\mu}\).

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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden
  2. 2.Department of Mathematics and Systems AnalysisAalto UniversityAaltoFinland
  3. 3.Dipartimento di Matematica e InformaticaUniversità di ParmaParmaItaly

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