Inverse Problem for Equations of Mixed Type with Lavrentev--Bitsadze Operator

I. A. Khadzhi

Matematičeskie zametki, Tome 91 (2012) no. 6, pp. 908-919

Résumé

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.

Détail

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