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Comparison results and improved quantified inequalities for semilinear elliptic equations

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Abstract

In this paper, we prove some pointwise comparison results between the solutions of some second-order semilinear elliptic equations in a domain \(\Omega \) of \(\mathbb {R}^n\) and the solutions of some radially symmetric equations in the equimeasurable ball \(\Omega ^*\). The coefficients of the symmetrized equations in \(\Omega ^*\) satisfy similar constraints as the original ones in \(\Omega \). We consider both the case of equations with linear growth in the gradient and the case of equations with at most quadratic growth in the gradient. Moreover, we show some improved quantified comparisons when the original domain is not a ball. The method is based on a symmetrization of the second-order terms.

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Acknowledgments

The authors thank the referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Centrale Marseille Institut de Mathématiques de Marseille, Aix Marseille Université, CNRS, UMR 7373, 13453, Marseille, France

    François Hamel

  2. Institut Fourier, Université Joseph Fourier, 100 rue des maths, BP 74, 38402, Saint-Martin d’Hères Cedex, France

    Emmanuel Russ

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  1. François Hamel
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Correspondence to François Hamel.

Additional information

This work has been carried out in the framework of the Labex Archimède (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government program managed by the French National Research Agency (ANR). The research leading to these results has received funding from the French ANR within the projects PREFERED and NONLOCAL (ANR-14-CE25-0013), and from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. 321186-ReaDi-Reaction-Diffusion Equations, Propagation and Modelling. Part of this work was also carried out during visits by the first author to the Departments of Mathematics of the University of California, Berkeley and of Stanford University, the hospitality of which is thankfully acknowledged.

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Hamel, F., Russ, E. Comparison results and improved quantified inequalities for semilinear elliptic equations. Math. Ann. 367, 311–372 (2017). https://doi.org/10.1007/s00208-016-1394-1

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