Geodesic
Inverse Problem for Equations of Mixed Type with Lavrentev–Bitsadze Operator
Matematičeskie zametki, Tome 91 (2012) no. 6, pp. 908-919 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For the equation of mixed elliptic-hyperbolic type $$ u_{xx}+(\operatorname{sgn}y)u_{yy}-b^2u=f(x) $$ in a rectangular domain $D=\{(x,y)\mid 0, where $\alpha$, $\beta$, and $b$ are given positive numbers, we study the problem with boundary conditions \begin{gather*} u(0,y)=u(1,y)=0,\qquad-\alpha\le y\le \beta, \\ u(x,\beta)=\varphi(x),\quad u(x,-\alpha)=\psi(x),\quad u_y(x,-\alpha)=g(x),\qquad 0\le x\le 1. \end{gather*} We establish a criterion for the uniqueness of the solution, which is constructed as the sum of the series in eigenfunctions of the corresponding eigenvalue problem and prove the stability of the solution.
Keywords: equation of mixed elliptic-hyperbolic type, inverse problem for partial differential equations, Lavrentev–Bitsadze operator, eigenvalue problem, stability of a solution, Cauchy–Bunyakovskii inequality.
Mots-clés : Weierstrass test for convergence 
@article{MZM_2012_91_6_a11,
     author = {I. A. Khadzhi},
     title = {Inverse {Problem} for {Equations} of {Mixed} {Type} with {Lavrentev{\textendash}Bitsadze} {Operator}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {908--919},
     year = {2012},
     volume = {91},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2012_91_6_a11/}
}
TY  - JOUR
AU  - I. A. Khadzhi
TI  - Inverse Problem for Equations of Mixed Type with Lavrentev–Bitsadze Operator
JO  - Matematičeskie zametki
PY  - 2012
SP  - 908
EP  - 919
VL  - 91
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/MZM_2012_91_6_a11/
LA  - ru
ID  - MZM_2012_91_6_a11
ER  - 
%0 Journal Article
%A I. A. Khadzhi
%T Inverse Problem for Equations of Mixed Type with Lavrentev–Bitsadze Operator
%J Matematičeskie zametki
%D 2012
%P 908-919
%V 91
%N 6
%U http://geodesic.mathdoc.fr/item/MZM_2012_91_6_a11/
%G ru
%F MZM_2012_91_6_a11
I. A. Khadzhi. Inverse Problem for Equations of Mixed Type with Lavrentev–Bitsadze Operator. Matematičeskie zametki, Tome 91 (2012) no. 6, pp. 908-919. http://geodesic.mathdoc.fr/item/MZM_2012_91_6_a11/

[1] A. M. Tikhonov, “Ob ustoichivosti obratnykh zadach”, DAN, 39:5 (1943), 195–198 | MR | Zbl

[2] M. M. Lavrentev, V. G. Romanov, S. T. Shishatskii, Nekorrektnye zadachi matematicheskoi fiziki i analiza, Nauka, M., 1980 | MR | Zbl

[3] V. G. Romanov, Nekotorye obratnye zadachi dlya uravneniya giperbolicheskogo tipa, Nauka, Novosibirsk, 1972 | MR | Zbl

[4] V. K. Ivanov, V. V. Vasin, V. P. Tanan, Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, Nauka, M., 1978 | MR | Zbl

[5] A. M. Denisov, Vvedenie v teoriyu obratnykh zadach, Izd-vo Mosk. un-ta, M., 1994

[6] K. B. Sabitov, “Kriterii edinstvennosti resheniya kraevoi zadachi dlya uravneniya smeshanno-sostavnogo tipa”, Trudy mezhdunarodnoi nauchnoi konferentsii “Differentsialnye uravneniya i smezhnye problemy”, posvyaschennoi yubileyu akademika V. A. Ilina, T. 2, Gilem, Ufa, 2008, 154–161

[7] K. B. Sabitov, “Ob odnoi kraevoi zadache dlya uravneniya smeshannogo tipa tretego poryadka”, Dokl. RAN, 427:5 (2009), 593–596 | MR | Zbl

[8] K. B. Sabitov, E. M. Safin, “Obratnaya zadacha dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa v pryamougolnoi oblasti”, Izv. vuzov. Matem., 2010, no. 4, 55–62 | MR | Zbl

[9] K. B. Sabitov, E. M. Safin, “Obratnaya zadacha dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa”, Matem. zametki, 87:6 (2010), 907–918 | MR

[10] K. B. Sabitov, I. A. Khadzhi, “Kraevaya zadacha dlya uravneniya Lavrenteva–Bitsadze s neizvestnoi pravoi chastyu”, Izv. vuzov. Matem., 2011, no. 5, 44–52 | Zbl

[11] M. M. Smirnov, Uravneniya smeshannogo tipa, Nauka, M., 1970 | MR | Zbl

[12] V. A. Ilin, “Edinstvennost i prinadlezhnost $W^1_2$ klassicheskogo resheniya smeshannoi zadachi dlya samosopryazhennogo giperbolicheskogo uravneniya”, Matem. zametki, 17:1 (1975), 91–101 | MR | Zbl

[13] V. A. Ilin, “Teorema o edinstvennosti i prinadlezhnosti klassu $W_2^1$ klassicheskogo resheniya smeshannoi zadachi dlya nesamosopryazhennogo giperbolicheskogo uravneniya v proizvolnom tsilindre”, Differents. uravneniya, 11:1 (1975), 60–65 | MR | Zbl