ISSN 1055-1344, Siberian Advances in Mathematics, 2011, Vol. 21, No. 4, pp. 262–273. c Allerton Press, Inc., 2011. On the Uniqueness of Generalized and Quasi-Regular Solutions to Equations of Mixed Type in R 3 T. D. Hristov 1** , N. I. Popivanov 1*** , and M. Schneider 2**** 1 University of Sofia, Sofia, Bulgaria 2 Karlsruhe Institute of Technology, Karlsruhe, Germany Received February 10, 2011 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. DOI: 10.3103/S1055134411040043 Keywords: mixed-type equation, boundary value problem, generalized solution, quasi-regular solution, uniqueness In memory of Professor Vladimir Vragov (1945-2002) 1. INTRODUCTION For m ∈ R,m ≥ 0, we study the equation L m [u]:= x m 3 [u x 1 x 1 + u x 2 x 2 ] − u x 3 x 3 + B 1 u x 1 + B 2 u x 2 + B 3 u x 3 + ru = x m 3 1 (u (1) ) + 1 2 u (1) ϕϕ − u (1) x 3 x 3 + a 1 u (1) + a 2 u (1) ϕ + a 3 u (1) x 3 + ru (1) = f (, ϕ, x 3 ) (1) and, for m ∈ R, 0 <m< 2, the equation ˜ L m [u] := u x 1 x 1 + u x 2 x 2 − (x m 3 u x 3 ) x 3 + ru = 1 (u (1) ) + 1 2 u (1) ϕϕ − (x m 3 u (1) x 3 ) x 3 + ru (1) = f (, ϕ, x 3 ) (2) written in Cartesian (x 1 ,x 2 ,x 3 ) or polar (, ϕ, x 3 ) coordinates, where x 1 = cos ϕ, x 2 = sin ϕ, u(x)= u(x 1 ,x 2 ,x 3 ) =: u (1) (, ϕ, x 3 ). The coefficients a 1 and a 2 depend on B 1 and B 2 in an obvious way (a 1 = B 1 cosϕ + B 2 sinϕ, a 2 = −1 (B 2 cosϕ − B 1 sinϕ)), and a 3 = B 3 . For m> 0, equation (1) is called of mixed type of the first kind (or Tricomi type) and equation (2) is called of mixed type of the second kind (or Keldysh type). We consider equation (1) in a simply connected region ∗ The text was submitted by the authors in English. ** E-mail: tsvetan@fmi.uni-sofia.bg *** E-mail: nedyu@fmi.uni-sofia.bg **** E-mail: manfred.schneider@math.uni-karlsruhe.de 262