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On the uniqueness of generalized and quasi-regular solutions to equations of mixed type in ℝ3

In memory of Professor Vladimir Vragov (1945–2002)

Abstract

Some three-dimensional (3D) problems for mixed type equations of first and second kind are studied. For equation of Tricomi type, they are 3D analogs of the Darboux (or Cauchy-Goursat) plane problem. Such type problems for a class of hyperbolic and weakly hyperbolic equations as well as for some hyperbolic-elliptic equations are formulated by M. Protter in 1952. In contrast to the well-posedness of the Darboux problem in the 2D case, the new 3D problems are strongly ill-posed. A similar statement of 3D problem for Keldysh-type equations is also given. For mixed type equations of Tricomi and Keldysh type, we introduce the notion of generalized or quasi-regular solutions and find sufficient conditions for the uniqueness of such solutions to the Protter’s problems. The dependence of lower order terms is also studied.

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Correspondence to T. D. Hristov.

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The text was submitted by the authors in English.

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Hristov, T.D., Popivanov, N.I. & Schneider, M. On the uniqueness of generalized and quasi-regular solutions to equations of mixed type in ℝ3 . Sib. Adv. Math. 21, 262–273 (2011). https://doi.org/10.3103/S1055134411040043

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Keywords

  • mixed-type equation
  • boundary value problem
  • generalized solution
  • quasi-regular solution
  • uniqueness