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Fractional differentiability for solutions of the inhomogeneous $ p$-Laplace system

Author: Michał Miśkiewicz
Journal: Proc. Amer. Math. Soc. 146 (2018), 3009-3017
MSC (2010): Primary 35B65, 35J92
Published electronically: February 28, 2018
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Abstract: It is shown that if $ p \ge 3$ and $ u \in W^{1,p}(\Omega ,\mathbb{R}^N)$ solves the inhomogeneous $ p$-Laplace system

$\displaystyle \operatorname {div} (\vert\nabla u\vert^{p-2} \nabla u) = f, \qquad f \in W^{1,p'}(\Omega ,\mathbb{R}^N), $

then locally the gradient $ \nabla u$ lies in the fractional Nikol'skiĭ space $ \mathcal {N}^{\theta ,2/\theta }$ with any $ \theta \in [ \tfrac {2}{p}, \tfrac {2}{p-1} )$. To the author's knowledge, this result is new even in the case of $ p$-harmonic functions, slightly improving known $ \mathcal {N}^{2/p,p}$ estimates. The method used here is an extension of the one used by A. Cellina in the case $ 2 \le p < 3$ to show $ W^{1,2}$ regularity.

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Michał Miśkiewicz
Affiliation: Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Keywords: $p$-Laplacian, degenerate elliptic systems, fractional order Nikol'ski{\u{\i}} spaces
Received by editor(s): August 2, 2017
Received by editor(s) in revised form: October 9, 2017
Published electronically: February 28, 2018
Additional Notes: The author’s research was supported by the NCN grant no. 2012/05/E/ST1/03232 (years 2013-2017).
Communicated by: Joachim Krieger
Article copyright: © Copyright 2018 American Mathematical Society

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