Abstract
In this paper, for the solution of the torsion problem about the equation Δu = −2 with homogeneous Dirichlet boundary conditions in a bounded convex domain in ℝn, we find a superharmonic function which implies the strict concavity of \({u^{{1 \over 2}}}\) and give some convexity estimates. It is a generalization of Makar-Limanov’s result (Makar-Limanov (1971)) and Ma-Shi-Ye’s result (Ma et al. (2012)).
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Acknowledgements
The second author was supported by National Key Research and Development Project (Grant No. SQ2020YFA070080) and National Natural Science Foundation of China (Grant Nos. 11871255 and 11721101). The third author was supported by National Natural Science Foundation of China (Grant Nos. 11971137 and 11771396). Part of this work was done while the second and third authors were visiting Department of Mathematics at The Chinese University of Hong Kong. They thank Professor Guohuan Qiu for his invitation and hospitality.
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Jia, X., Ma, XN. & Shi, S. Remarks on convexity estimates for solutions of the torsion problem. Sci. China Math. 66, 1003–1020 (2023). https://doi.org/10.1007/s11425-021-1957-7
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DOI: https://doi.org/10.1007/s11425-021-1957-7