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

In this paper we study the following torsion problem { −∆u = 1 in Ω, u = 0 on ∂Ω. Let Ω ⊂ R be a bounded, convex domain and u0(x) be the solution of above problem with its maximum y0 ∈ Ω. Steinerberger [14] proved that there are universal constants c1, c2 > 0 satisfying λmax ( Du0(y0) ) ≤ −c1exp ( −c2 diam(Ω) inrad(Ω) )

[1]  A note on the critical points of the torsion function , 2019, 1907.08376.

[2]  X. Cabré,et al.  Stable solutions of semilinear elliptic problems in convex domains , 1998 .

[3]  E. Lieb,et al.  On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation , 1976 .

[4]  R. Magnanini.  An introduction to the study of critical points of solutions of elliptic and parabolic equations , 2016, 1604.00530.

[5]  Bernhard Kawohl,et al.  Rearrangements and Convexity of Level Sets in PDE , 1985 .

[6]  B. Gidas,et al.  Symmetry and related properties via the maximum principle , 1979 .

[7]  M. Grossi.  A Morse Lemma for Degenerate Critical Points of Solutions of Nonlinear Equations in ℝ2 , 2018, Advanced Nonlinear Studies.

[8]  Thomas Beck.  Uniform Level Set Estimates for Ground State Eigenfunctions , 2018, SIAM J. Math. Anal..

[9]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[10]  S. Steinerberger.  Topological Bounds for Fourier Coefficients and Applications to Torsion , 2017, 1705.02651.

[11]  Catherine Bandle,et al.  Harmonic Radius and Concentration of Energy; Hyperbolic Radius and Liouville's Equations Δ U = eU and Δ U = Un+2/n-2 , 1996, SIAM Rev..

[12]  ON THE NUMBER OF CRITICAL POINTS OF SOLUTIONS OF SEMILINEAR EQUATIONS IN R , 2019 .

[13]  N. Nadirashvili,et al.  Convexity of level sets for elliptic problems in convex domains or convex rings: Two counterexamples , 2013, 1304.3355.