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Regularity results for a priori bounded minimizers of non-autonomous functionals with discontinuous coefficients

  • Raffaella Giova und Antonia Passarelli di Napoli EMAIL logo
Veröffentlicht/Copyright: 20. Juni 2017

Abstract

We prove the higher differentiability and the higher integrability of the a priori bounded local minimizers of integral functionals of the form

(v,Ω)=Ωf(x,Dv(x))dx,

with convex integrand satisfying p-growth conditions with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x-variable belongs to a suitable Sobolev space. The a priori boundedness of the minimizers allows us to obtain the higher differentiability under a Sobolev assumption which is independent on the dimension n and that, in the case pn-2, improves previous known results. We also deal with solutions of elliptic systems with discontinuous coefficients under the so-called Uhlenbeck structure. In this case, it is well known that the solutions are locally bounded and therefore we obtain analogous regularity results without the a priori boundedness assumption.

MSC 2010: 49N60; 35J60; 49N99

Communicated by Frank Duzaar


Funding statement: The authors have partly been supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The first author has been partially supported by Università degli Studi di Napoli “Parthenope” through the Project “Sostegno alla ricerca individuale (2015–2017)”.

Acknowledgements

The authors thanks the anonymous referee for the careful reading of the manuscript and for valuable comments.

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Received: 2016-12-02
Revised: 2017-04-12
Accepted: 2017-06-12
Published Online: 2017-06-20
Published in Print: 2019-01-01

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