Mathematical Journal of Ibaraki University

A relation between the Lorentzian inversion and the Bateman transformation

Katsunori Shimomura

2012 年 44 巻 p. 1-5
発行日: 2012/05/01
公開日: 2013/04/01

DOIhttps://doi.org/10.5036/mjiu.44.1

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In our previous paper [1], 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.

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On generalized Kronecker function rings, II

Akira Okabe , Ryûki Matsuda

2012 年 44 巻 p. 7-16
発行日: 2012/05/01
公開日: 2013/04/01

DOIhttps://doi.org/10.5036/mjiu.44.7

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We define a Kronecker function ring for any L-semistar operation, and study ascents and descents of L-semistar operations.

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On the weighted rearrangement of functions and degenerate nonlinear elliptic equations

Hiroshi Ando , Toshio Horiuchi

2012 年 44 巻 p. 17-31
発行日: 2012/05/01
公開日: 2013/04/01

DOIhttps://doi.org/10.5036/mjiu.44.17

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Let Ω be a bounded domain of Rⁿ. We shall deal with boundary value problems of the following form −∑_i=1ⁿ ∂ ∕ ∂x_i(|x|^α a_i(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.

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