The hot spots conjecture can be false: some numerical examples

The hot spots conjecture is only known to be true for special geometries. It can be shown numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Additionally, it can be shown numerically that the ratio between the maximal/minimal value inside the domain and its maximal/minimal value on the boundary can be larger than 1+ 10. Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well. AMS classification scheme numbers: 35J25, 35P20, 65F15, 65M38, 78A46

[1]  Chia-Ming Fan,et al.  The method of fundamental solutions for eigenproblems in domains with and without interior holes , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[2]  The “hot spots” conjecture for a certain class of planar convex domains , 2009 .

[3]  E. O. Asante-Asamani,et al.  A second-order exponential time differencing scheme for non-linear reaction-diffusion systems with dimensional splitting , 2020, J. Comput. Phys..

[4]  Fioralba Cakoni,et al.  A boundary integral equation method for the transmission eigenvalue problem , 2017 .

[5]  Chris Judge,et al.  Euclidean triangles have no hot spots , 2018, Annals of Mathematics.

[6]  Barry Simon,et al.  The essential spectrum of Neumann Laplacians on some bounded singular domains , 1991 .

[7]  Andreas Kleefeld,et al.  A Global Galerkin Method for Solving the Exterior Neumann Problem for the Helmholtz Equation Using Panich's Integral Equation Approach , 2013, SIAM J. Sci. Comput..

[8]  R. Atar.  Invariant Wedges for a Two-Point Reflecting Brownian Motion and the ``Hot Spots'' Problem , 2001 .

[9]  Richard F. Bass,et al.  Fiber Brownian motion and the `hot spots''problem Duke Math , 2000 .

[10]  B. Siudeja,et al.  Hot spots conjecture for a class of acute triangles , 2015 .

[11]  W. Beyn.  An integral method for solving nonlinear eigenvalue problems , 2012 .

[12]  A. Seybert,et al.  An advanced computational method for radiation and scattering of acoustic waves in three dimensions , 1985 .

[13]  D. Krejčiřík,et al.  Location of hot spots in thin curved strips , 2017, Journal of Differential Equations.

[14]  Stefan Steinerberger,et al.  Hot spots in convex domains are in the tips (up to an inradius) , 2019, Communications in Partial Differential Equations.

[15]  R. Hiptmair,et al.  Boundary Element Methods , 2021, Oberwolfach Reports.

[16]  K. Burdzy,et al.  On the “Hot Spots” Conjecture of J. Rauch , 1999 .

[17]  K. Atkinson.  The Numerical Solution of Integral Equations of the Second Kind , 1997 .

[18]  A. Kleefeld,et al.  The method of fundamental solutions for computing acoustic interior transmission eigenvalues , 2018 .

[19]  M. Pascu.  Scaling coupling of reflecting Brownian motions and the hot spots problem , 2002 .

[20]  J. Rauch.  Lecture #1. Five problems: An introduction to the qualitative theory of partial differential equations , 1975 .

[21]  Fikret Er,et al.  Hot spots. , 2014, CJEM.

[22]  Krzysztof Burdzy.  The hot spots problem in planar domains with one hole , 2004 .

[24]  Jeffrey C. Lagarias,et al.  Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions , 1998, SIAM J. Optim..

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

[26]  A. Kleefeld,et al.  BOUNDARY ELEMENT COLLOCATION METHOD FOR SOLVING THE EXTERIOR NEUMANN PROBLEM FOR HELMHOLTZ’S EQUATION IN THREE DIMENSIONS , 2022 .

[27]  Shape Optimization for Interior Neumann and Transmission Eigenvalues , 2018, Integral Methods in Science and Engineering.

[28]  L. Shampine.  Vectorized adaptive quadrature in MATLAB , 2008 .

[29]  Closed nodal lines and interior hot spots of the second eigenfunction of the Laplacian on surfaces , 2000, math/0009153.

[30]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[31]  Denis S. Grebenkov,et al.  Geometrical Structure of Laplacian Eigenfunctions , 2012, SIAM Rev..

[32]  Bo Zhang,et al.  The Factorization Method for Reconstructing a Penetrable Obstacle with Unknown Buried Objects , 2013, SIAM J. Appl. Math..

[33]  New Numerical Results for the Optimization of Neumann Eigenvalues , 2020 .

[34]  A planar convex domain with many isolated “ hot spots” on the boundary , 2013 .

[35]  R. Atar,et al.  On Neumann eigenfunctions in lip domains , 2004 .

[36]  David Jerison,et al.  The “hot spots” conjecture for domains with two axes of symmetry , 2000 .

[37]  Olaf Steinbach,et al.  Convergence Analysis of a Galerkin Boundary Element Method for the Dirichlet Laplacian Eigenvalue Problem , 2012, SIAM J. Numer. Anal..

[38]  Simon R. Arridge,et al.  Solving Boundary Integral Problems with BEM++ , 2015, ACM Trans. Math. Softw..

[39]  Extreme Values of the Fiedler Vector on Trees , 2019, ArXiv.

[40]  Wendelin Werner,et al.  A counterexample to the “hot spots” conjecture , 1998 .