The hot spots conjecture can be false: Some numerical examples

Understand

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.