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Regularity in oscillatory nonlinear elliptic systems

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Abstract

We give uniform bounds in Morrey–Sobolev spaces H 1,μ for weak solutions (u ε ) to oscillatory nonlinear elliptic systems. The main novelty is in the precise qualitative description of the ε-uniform Morrey exponent μ in terms of the ellipticity ratio of the elliptic systems. This is achieved by showing that weak solutions of general nonlinear elliptic systems with Lipschitz coefficients are of class W 2,p (locally or globally depending on assumptions) for an exponent p > 2 that is independent of dimensions and for which p − 2 is inversely proportional to the ellipticity ratio.

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  1. Mathematical Institute, 24–29 St Giles’, Oxford, OX1 3LB, UK

    Jan Kristensen & Christof Melcher

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  1. Jan Kristensen
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  2. Christof Melcher
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Correspondence to Christof Melcher.

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Kristensen, J., Melcher, C. Regularity in oscillatory nonlinear elliptic systems. Math. Z. 260, 813–847 (2008). https://doi.org/10.1007/s00209-008-0302-0

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