Geodesic
Proof of uniqueness and membership in $W^1_2$ of the classical solution of a mixed problem for a self-adjoint hyperbolic equation
Matematičeskie zametki, Tome 17 (1975) no. 1, pp. 91-101 
V. A. Il'in. Proof of uniqueness and membership in $W^1_2$ of the classical solution of a mixed problem for a self-adjoint hyperbolic equation. Matematičeskie zametki, Tome 17 (1975) no. 1, pp. 91-101. http://geodesic.mathdoc.fr/item/MZM_1975_17_1_a10/
@article{MZM_1975_17_1_a10,
     author = {V. A. Il'in},
     title = {Proof of uniqueness and membership in $W^1_2$ of the classical solution of a mixed problem for a self-adjoint hyperbolic equation},
     journal = {Matemati\v{c}eskie zametki},
     pages = {91--101},
     year = {1975},
     volume = {17},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_17_1_a10/}
}
TY  - JOUR
AU  - V. A. Il'in
TI  - Proof of uniqueness and membership in $W^1_2$ of the classical solution of a mixed problem for a self-adjoint hyperbolic equation
JO  - Matematičeskie zametki
PY  - 1975
SP  - 91
EP  - 101
VL  - 17
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MZM_1975_17_1_a10/
LA  - ru
ID  - MZM_1975_17_1_a10
ER  - 
%0 Journal Article
%A V. A. Il'in
%T Proof of uniqueness and membership in $W^1_2$ of the classical solution of a mixed problem for a self-adjoint hyperbolic equation
%J Matematičeskie zametki
%D 1975
%P 91-101
%V 17
%N 1
%U http://geodesic.mathdoc.fr/item/MZM_1975_17_1_a10/
%G ru
%F MZM_1975_17_1_a10

Voir la notice de l'article provenant de la source Math-Net.Ru

In this article a uniqueness theorem for the classical solution of a mixed problem is proved under minimal assumptions on the coefficients of the differential operator for admitting the Fourier method of a hyperbolic second-order equation in an $(N+1)$-dimensional cylinder, whose cross section is a completely arbitrary bounded $N$-dimensional domain. Furthermore, it is proved that the classical solution of the indicated mixed problem, whenever it exists, belongs to the class $W^1_2$ and is the generalized solution from $W^1_2$ of the same problem.