Bateman's transformation is associated with the Lorentzian metric and preserves solutions of the wave equation. We generalize Bateman's transformation for general indefinite semi-euclidean metrics. Then we show that the generalized transformation preserves solutions of the equation associated with given indefinite metric.
A simple constrained minimization problem with an integral constraint describes a symmetry breaking of a circular front around a point source. As a single control parameter, the total flux φ from the source, is varied, apparently polygonal solutions with an arbitrary number of corners m are shown to bifurcate from the circular solution. Our asymptotic analysis shows that the branches with m ≥ 3 bifurcate supercritically at φ = π(m2 + 2) and continue as φ → ∞ whereas those with m = 1 or 2 bifurcate subcritically and are terminated at φ = 3π and 4π, respectively. The second variation can be evaluated directly for the circular state which is proven to be the minimizing solution only up to φ = 3π.
In this paper we give an alternative proof of the Ohsawa-Takegoshi extension theorem. We prove the theorem using three weight functions which Hörmander used to obtain L2 estimates for solutions of the ∂ problem in pseudoconvex domains.