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Boundedness and differentiability for nonlinear elliptic systems


Author: Jana Björn
Journal: Trans. Amer. Math. Soc. 353 (2001), 4545-4565
MSC (2000): Primary 35J70; Secondary 35B35, 35B65, 35D10, 35J60, 35J85
DOI: https://doi.org/10.1090/S0002-9947-01-02834-3
Published electronically: May 9, 2001
MathSciNet review: 1851183
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Abstract: We consider the elliptic system $\operatorname{div} (\mathcal{A}^j (x,u,\nabla u)) = \mathcal{B}^j (x,u,\nabla u)$, $j=1,\ldots,N,$and an obstacle problem for a similar system of variational inequalities. The functions $\mathcal{A}^j$ and $\mathcal{B}^j$ satisfy certain ellipticity and boundedness conditions with a $p$-admissible weight $w$ and exponent $1<p\le2$. The growth of $\mathcal{B}^j$ in $\vert\nabla u\vert$ and $\vert u\vert$ is of order $p-1$. We show that weak solutions of the above systems are locally bounded and differentiable almost everywhere in the classical sense.


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Additional Information

Jana Björn
Affiliation: Department of Mathematics, Lund Institute of Technology, P. O. Box 118, SE-221 00 Lund, Sweden
Email: jabjo@maths.lth.se, jabjo@mai.liu.se

DOI: https://doi.org/10.1090/S0002-9947-01-02834-3
Keywords: Elliptic system, $p$-admissible weight, obstacle problem, local boundedness, differentiability a.e.
Received by editor(s): December 8, 1999
Received by editor(s) in revised form: November 20, 2000
Published electronically: May 9, 2001
Additional Notes: The results of this paper were obtained while the author was visiting the University of Michigan, Ann Arbor, on leave from the Linköping University. The research was supported by grants from the Swedish Natural Science Research Council, the Knut and Alice Wallenberg Foundation and Gustaf Sigurd Magnusons fond of the Royal Swedish Academy of Sciences.
Article copyright: © Copyright 2001 American Mathematical Society

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