Geodesic
Boundary problem for Lavrent'ev--Bitsadze equation with two internal lines of change of a~type
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2016), pp. 23-37

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We study the problem with boundary conditions of the first and second kind on the boundary of the rectangular area for an equation with two internal perpendicular lines of change of a type. With the use of spectral method we prove the uniqueness and the existence of a solution. Obtained in the process of separation of variables, the eigenvalue problem for an ordinary differential equation is not self-adjoint, and the system of root functions is not orthogonal. We construct corresponding biorthogonal system of functions and prove its completeness, based on which we establish a criterion for the uniqueness of the problem. A solution to the problem is constructed as a sum of biorthogonal series.
Keywords: mixed type equation, mixed boundary-value problem, biorthogonal system functions, completeness, existence and uniqueness of solution. 
@article{IVM_2016_3_a2,
     author = {A. A. Gimaltdinova and K. V. Kurman},
     title = {Boundary problem for {Lavrent'ev--Bitsadze} equation with two internal lines of change of a~type},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {23--37},
     publisher = {mathdoc},
     number = {3},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2016_3_a2/}
}
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A. A. Gimaltdinova; K. V. Kurman. Boundary problem for Lavrent'ev--Bitsadze equation with two internal lines of change of a~type. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2016), pp. 23-37. http://geodesic.mathdoc.fr/item/IVM_2016_3_a2/