Torsion and ground state maxima: close but not the same

Could the location of the maximum point for a positive solution of a semilinear Poisson equation on a convex domain be independent of the form of the nonlinearity? Cima and Derrick found certain evidence for this surprising conjecture. We construct counterexamples on the half-disk, by working with the torsion function and first Dirichlet eigenfunction. On an isosceles right triangle the conjecture fails again. Yet the conjecture has merit, since the maxima of the torsion function and eigenfunction are unexpectedly close together. It is an open problem to quantify this closeness in terms of the domain and the nonlinearity

Properties of heat kernels

We prove two kinds of results related to the asymptotic behavior of the Dirichlet or Neumann heat kernels. The first concerns the monotonicity of the Neumann heat kernel toward the boundary. We show that the diagonal of the radial Neumann heat kernel of the ball is increasing toward the boundary if and only if the dimension of the space is bigger than two. The second problem is to give good bounds for the first few eigenvalues of the Dirichlet Laplacian on bounded sets. This gives long time asymptotics of the Dirichlet heat kernel. We prove new upper and lower bounds for triangular domains involving various geometric measurements of the domain