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Journal of Mathematical Sciences, Vol. 274, No. 2, August, 2023
BITSADZE–SAMARSKII TYPE PROBLEM FOR A
MIXED TYPE EQUATION THAT IS ELLIPTIC IN
THE FIRST QUADRANT OF THE PLANE
Rakhimjon Zunnunov
Romanovsky Institute of Mathematics
Academy of Sciences of the Republic of Uzbekistan
9, University St., Tashkent 100174, Uzbekistan
zunnunov@mail.ru
We consider the problem of Bitsadze–Samarskii type for a generalized Tricomi equation
with a spectral parameter in the case where the equation is elliptic in the first quadrant
of the plane. We establish the existence and uniqueness of a solution to the problem.
Bibliography:5titles.
1 Statement of the Problem
We consider the equation
sgn y|y|muxx +uyy −λ2|y|mu=0 (1.1)
in the unbounded domain Ω = Ω1∪l1∪Ω2,whereΩ
1={(x, y):x>0,y > 0}and Ω2is a
domain in the half-plane y<0 bounded by the segment AB of the straight line y=0andthe
characteristics
AC : x−[2/(m+2)](−y)(m+2)/2=0,
BC : x+[2/(m+2)](−y)(m+2)/2=1
of Equation (1.1) outgoing from the points A(0,0) and B(1,0). We set β=m/(2m+4),
l1={(x, y): 0<x<1,y =0},l2={(x, y): x>1,y =0},l3={(x, y): y>0,x =0},
θ0(x)=(x/2,−(m+2)/2·x/2] 2
m+2 ) is the point of intersection of the characteristic of Equation
(1.1) outgoing from the point (x, 0) ∈l1with the characteristic AC. We assume that m, λ ∈R,
m= const >0, and
λ=⎧
⎨
⎩
λ1,y>0,
λ2,y<0.
Problem BS ∞.Find a function u(x, y) such that
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301
DOI 10.1007/s10958-023-06597-6
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