Abstract
We prove a comparison theorem for super- and sub-solutions with non-vanishing gradients to semilinear PDEs provided a nonlinearity f is Lp function with p > 1. The proof is based on a strong maximum principle for solutions of divergence type elliptic equations with VMO leading coefficients and with lower order coefficients from a Kato class. An application to estimation of periodic water waves profiles is given.
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Acknowledgements
V. K. acknowledges the support of the Swedish Research Council (VR) Grant EO418401. A. N. was partially supported by Russian Foundation for Basic Research, Grant 18-01-00472. He also thanks the Linköping University for the hospitality during his visit in January 2018.
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Kozlov, V., Nazarov, A. A Comparison Theorem for Nonsmooth Nonlinear Operators. Potential Anal 54, 471–481 (2021). https://doi.org/10.1007/s11118-020-09834-8
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DOI: https://doi.org/10.1007/s11118-020-09834-8
Keywords
- Semi-linear elliptic equation
- Non-smooth nonlinearity
- Comparison principal
- VMO coefficients
- Kato classes
- Strong maximum principle