In our previous paper , we proved that every transformation which preserves the wave equation is a similarity or a Lorentzian inversion composed with similarities or a Bateman transformation composed with similarities. In this paper, we give several relations between Bateman transformation and Lorentzian inversion. We also prove that only Lorentzian inversion or Bateman transformation is enough to generate the set of all transformations which preserve the wave equation.
Let Ω be a bounded domain of Rn. We shall deal with boundary value problems of the following form −∑i=1n ∂ ∕ ∂xi (|x|α ai(x, u, ∇u)) = |x|α H(x, u) in Ω, u = g on ∂Ω. Here α > 1 - n, u is the relevant solution, ∇u is its gradient and H is a given real-valued function. Under proper assumptions a pri-ori estimates of solutions u to the problem (0.1) are established by virtue of weighted rearrangement of functions and weighted isoperimetric inequalities.