Mikheil Usanetashvili
abstract:
The first boundary value problem is studied for
second order general elliptic equations degenerating on the whole boundary.
In accordance with the type of degeneration, the cases are distinguished where
the whole boundary becomes free of boundary conditions. For a class
of second order degenerating elliptic equations, a new approach is proposed
which enables one to prove the correctness of the Dirichlet problem.
For second order general elliptic equations degenerating on a part of
the boundary, conditions are found guaranteeing the correctness of the problem
with oblique derivative. For the solution of this problem, an a priori estimate
is obtained. A boundary value problem of conjugation type is studied in
weighted spaces for a class of degenerating second order hyperbolic systems
with discontinuous coefficients. The problems with oblique derivative
are also investigated for mixed type equations with a Lavrent'ev--Bitsadze
operator as the principal part.
Mathematics Subject Classification: 35J70, 35L80, 35M05.
Key words and phrases: Degenerating elliptic, hyperbolic and mixed type equations, the Dirichlet problem, problem with oblique derivative, extremum principle, index of the problem.