The spectral gap to torsion problem for some non-convex domains

作者：Chen, Hua^*; Luo, Peng

来源：Calculus of Variations and Partial Differential Equations, 2021, 60(6): 211.

DOI：10.1007/s00526-021-02080-9

• LEVEL SETS
• CONVEXITY

In this paper we study the following torsion problem @@@ {-Delta u = 1 in Omega, u = 0 on partial derivative Omega, @@@ Let Omega subset of R-2 be a bounded, convex domain and u(0)(x) be the solution of above problem with its maximum y(0) is an element of Omega. Steinerberger (J Funct Anal 274:1611-1630, 2018) proved that there are universal constants c(1), c(2) > 0 satisfying @@@ lambda(max) (D(2)u(0)(y(0))) <= -c(1)exp (-c(2) diam(Omega)/inrad(Omega)). @@@ 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(2)u(0)(y(0)) for some non-convex smooth bounded domains, which gives a negative answer to above open problem.

• 单位
  武汉大学