We study the semilinear Poisson equation Δu=f(x,u)inB1. (1) Our main results provide conditions on f which ensure that weak solutions of (1) belong to C1,1(B1/2). In some configurations, the conditions are sharp.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledgements: We thank Henrik Shahgholian for introducing us to the regularity problem for semilinear equations. Special thanks go to John Andersson for valuable feedback on a preliminary version of the paper. E. Indrei acknowledges: (i) support from NSF Grants OISE-0967140 (PIRE), DMS-0405343, and DMS-0635983 administered by the Center for Nonlinear Analysis at Carnegie Mellon University and an AMS-Simons Travel Grant; (ii) the hospitality of the Max Planck Institute in Leipzig and University of Oxford where part of the research was carried out. L. Nurbekyan was partially supported by KAUST baseline and start-up funds and KAUST SRI, Uncertainty Quantification Center in Computational Science and Engineering.