Abstract
We establish the \({C^{1,\alpha}}\) partial regularity of vectorial minimizers of non autonomous convex integral functionals of the type
with p-growth into the gradient variable. As a novel feature, we allow discontinuous dependence on the x variable, through a suitable Sobolev function. The Hölder’s continuity of the gradient of the minimizers is obtained outside a negligible set and this an unavoidable feature in the vectorial setting. Here, the so called singular set has to take into account also of the possible discontinuity of the coefficients.
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25 April 2020
The author has retracted this article [1]. The proof of Theorem 1 contains a mistake that can���t be corrected with an erratum.
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To the memory of my father
This work has been partially supported by INdAM-GNAMPA Project—“Stime quantitative in disuguaglianze geometriche” (2014) and by the Project “Metodi matematici per la modellizzazione di problemi di localizzazione e di trasporto ottimo” (Legge5/2009 Regione Campania).
The author has retracted this article. The proof of Theorem 1 contains a mistake that can’t be corrected with an erratum. In equation (4.1) it is assumed that the average of |k|n over a ball BR(x0) is bounded by a constant and this doesn’t imply that the average remains bounded for every smaller concentric ball. This boundedness at every smaller scale is necessary to perform the iteration procedure. Assuming that such boundedness holds at every scale would lead to a singular set of full measure but whose complementary is not necessarily open and this would prevent the use of the characterization of the Holder’s continuous functions by Campanato.
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Passarelli di Napoli, A. RETRACTED ARTICLE: A \({C^{1,\alpha}}\) partial regularity result for non-autonomous convex integrals with discontinuous coefficients. Nonlinear Differ. Equ. Appl. 22, 1319–1343 (2015). https://doi.org/10.1007/s00030-015-0324-3
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DOI: https://doi.org/10.1007/s00030-015-0324-3