In 1955 and 1956 Payne, Pólya, and Weinberger considered the problem of bounding ratios of eigenvalues for homogeneous membranes of arbitrary shape [PPW1, PPW2]. Among other things, they showed that… (More)

For a domain Ω contained in a hemisphere of the n–dimensional sphere Sn we prove the optimal result λ2/λ1(Ω) ≤ λ2/λ1(Ω ⋆) for the ratio of its first two Dirichlet eigenvalues where Ω⋆, the symmetric… (More)

To determine the sharp constants for the one dimensional Lieb– Thirring inequalities with exponent γ ∈ (1/2, 3/2) is still an open problem. According to a conjecture by Lieb and Thirring the sharp… (More)

It is shown that the sharp constant in the Hardy-Sobolev-Maz’ya inequality on the upper half space H3 ⊂ R3 is given by the Sobolev constant. This is achieved by a duality argument relating the… (More)

We give an explicit formula for the change of speed of pushed and bistable fronts of the reaction-diffusion equation when a small cutoff is applied to the reaction term at the unstable or metastable… (More)

We prove the optimal lower bound ¿2-^1 > 3n2/d2 for the difference of the first two eigenvalues of a one-dimensional Schrödinger operator -d2/dx2 + V'x) with a symmetric single-well potential on an… (More)

We study the minimal speed of propagating fronts of convection-reaction-diffusion equations of the form u(t)+microphi(u)u(x)=u(xx)+f(u) for positive reaction terms with f(')(0)>0. The function phi(u)… (More)

We give an integral variational characterization for the speed of fronts of the nonlinear diffusion equation ut = uxx + f(u) with f(0) = f(1) = 0, and f > 0 in (0, 1), which permits, in principle,… (More)