Solvability for Couple Elliptic Partial Differential Equations

As it famous, the solvability of any mathematical problem play an essential role by many investigators whose interest in the numerical solution for such mathematical problems. Because it gives these investigators the green light about the ability to solve such problems numerically. Various types of real life issues are described mathematically either by ODEs or by PDEs. The solvability for these types of problems (ODEs and PDEs) are studied usually in finite dimensional space, and are studied in infinite dimensional space (Hilbert space) rarely. In the recent years, the interesting for studying more general type of PDEs (system of PDEs) began arise, especially in infinite dimensional spaces. For these reasons, this work is devoted to study the solvability (in infinite dimension) of couple elliptic partial differential equations (CEPDEs) with four different types of boundary conditions (BCs): Dirichlet (DBCs), Neumann(NBCs), Robin(RBCs) and Mixed BCs(MBCs). The weak formulation (WF) for each problem according to the type of each BC is found. The existence and the uniqueness of the “couple” solution for the issues proposed (the CEPDEs with every kind of BCs) is proved by employing the LaxـMilgram theorem (LMTH) by using the Poincare-Friedrichs inequality (PFI) in the cases of homogenous BCs (HBCs) and through using the generalization PFI (GPFI) and the trace Operator (TRO) in the cases of nonhomogeneous BCs(NHBCs). The results were obtained in this work give the green light to the researchers about seeking the numerical solution for these types of problems.