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On two boundary value problems for mixed type equations with perpendicular lines of type change. (Russian) Zbl 1022.35029

The authors study boundary value problems for the following partial differential equation of mixed type \[ 0 = \begin{cases} u_{xx} - u_y + f(x,y) &\quad \text{in } \Omega_0 \\ u_{xx} - u_{yy} - \lambda_i^2\operatorname{sign} yu &\quad\text{in } \Omega_i, \end{cases} \] in a complex domain \(\Omega = \Omega_0\cup \Omega_i\). The equation is supplemented by the so-called nonlocal boundary conditions.
The purpose of the article is to prove that the boundary value problem admits a unique regular solution. The proof is based on reducing the boundary value problem under consideration to a Volterra integral equation of the second order.

MSC:

35M10 PDEs of mixed type
45D05 Volterra integral equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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