Geodesic
On periodic in the plane solutions of second order linear hyperbolic systems
Archivum mathematicum, Tome 33 (1997) no. 4, pp. 253-272 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Sufficient conditions for the problem \[ {\partial ^2 u\over \partial x\partial y}=P_0(x,y)u+ P_1(x,y){\partial u\over \partial x}+P_2(x,y){\partial u\over \partial y}+ q(x,y), u(x+\omega _1,y)=u(x,y),\quad u(x,y+\omega _2)=u(x,y) \] to have the Fredholm property and to be uniquely solvable are established, where $\omega _1$ and $\omega _2$ are positive constants and $P_j:R^2\rightarrow R^{n\times n}$ $(j=0,1,2)$ and $q:R^2\rightarrow R^n$ are continuous matrix and vector functions periodic in $x$ and $y$.
Sufficient conditions for the problem \[ {\partial ^2 u\over \partial x\partial y}=P_0(x,y)u+ P_1(x,y){\partial u\over \partial x}+P_2(x,y){\partial u\over \partial y}+ q(x,y), u(x+\omega _1,y)=u(x,y),\quad u(x,y+\omega _2)=u(x,y) \] to have the Fredholm property and to be uniquely solvable are established, where $\omega _1$ and $\omega _2$ are positive constants and $P_j:R^2\rightarrow R^{n\times n}$ $(j=0,1,2)$ and $q:R^2\rightarrow R^n$ are continuous matrix and vector functions periodic in $x$ and $y$.
Classification : 35B10, 35L10, 35L20, 35L55
Keywords: hyperbolic system; periodic solution; F property 
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Kiguradze, Tariel. On periodic in the plane solutions of second order linear hyperbolic systems. Archivum mathematicum, Tome 33 (1997) no. 4, pp. 253-272. http://geodesic.mathdoc.fr/item/ARM_1997_33_4_a0/

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