Abstract
In this paper we study the following torsion problem
Let \(\Omega \subset \mathbb R^2\) be a bounded, convex domain and \(u_0(x)\) be the solution of above problem with its maximum \(y_0\in \Omega \). Steinerberger (J Funct Anal 274:1611–1630, 2018) proved that there are universal constants \(c_1, c_2>0\) satisfying
And in Steinerberger (2018) he proposed following open problem: “Does above result hold true on domains that are not convex but merely simply connected or perhaps only bounded? The proof uses convexity of the domain \(\Omega \) in a very essential way and it is not clear to us whether the statement remains valid in other settings”. Here by some new idea involving the computations on Green’s function, we compute the spectral gap \(\lambda _{\max }D^2u_0(y_0)\) for some non-convex smooth bounded domains, which gives a negative answer to above open problem.
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Acknowledgements
Part of this work was done while Peng Luo was visiting the Mathematics Department of the University of Rome “La Sapienza” whose members he would like to thank for their warm hospitality. Also the authors would like to thank Prof. Guofang Wang for his helpful discussions and suggestions. Hua Chen was supported by NSFC Grants (Nos. 11631011, 11626251). Peng Luo was supported by NSFC Grants (Nos. 11701204, 11831009) and the Fundamental Research Funds for the Central Universities (No. KJ02072020-0319).
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Communicated by M. del Pino.
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