References for existence of solutions to overdetermined system of partial differential inequalities I want to show the existence of solutions to an overdetermined system of second-order partial differential inequalities in a given region $\Omega\subset\mathbb{R}^2$, 
\begin{equation*}
\begin{cases}
P_1(u)\geq 0,&\\
P_2(u)\geq 0,&\\
P_3(u)\geq 0.&
\end{cases}
\end{equation*}
Updated: $P_i$ are linear/quasilinear/nonlinear second-order differential operators, 
\begin{equation*}
\begin{cases}
P_1(u)=&\hspace{-0.3cm}xu_{xx}+yu_{xy}+x^2u_{yy}+4u_x.\\
P_2(u)=&\hspace{-0.3cm}u_{xx}u_{yy}-u_{xy}^2+(xu_x-yu_y)u_{xx}+(yu_x-x^2u_y)u_{xy}+x(xu_x-yu_y)u_{yy}.\\
P_3(u)=&\hspace{-0.3cm}(u_{xx}u_{yy}-u_{xy}^2)u_x+(xu_x^2-yu_xu_y-x^2u_y^2)u_{xx}+(yu_x^2-x^2u_xu_y-xyu_y^2)u_{xy}+(x^2u_x^2-xyu_xu_y-y^2u_y^2)u_{yy}.
\end{cases}
\end{equation*}
To do this, I am trying to solve for $(u,f_1,f_2,f_3)$ of the following system of PDEs,
\begin{equation*}
\begin{cases}
P_1(u)=f_1^2,&\\
P_2(u)=f_2^2,&\\
P_3(u)=f_3^2.&
\end{cases}
\end{equation*}
Could anyone suggest any references to deal with either system? 
Any suggestion is appreciated.
Updated: Is it true that if I find all compatible conditions for $f_1,f_2,f_3$ and they are satisfied, then this system has a solution?
 A: Existence theorems for real analytic pde systems were developed by Riquier and Janet in the first quarter of the twentieth century (see, for example, Riquier's 1910 book Les systèmes d'équations aux dérivées partielles).
An up to date treatment of their work, its modernisation by Pommaret, and its subsequent extensions is given in Seiler's book Involution The formal theory of differential equations and its applications in computer algebra. A lot of progress in 100 years.
Edit: an application of these ideas to overdetermined systems in particular can be found in Kunio Kakié "On Regularity of Solutions to Overdetermined Non Linear Partial Differential Equations", Commetarii Mathematici Universitatis Sancti Pauli Vol. 52, No. 2 2003, pp. 125-138. Note that as per the comments added below, this paper discusses local properties only.
Edit(2): regarding your question about compatibility conditions for the $f$ functions, the answer is "highly likely, but not guaranteed". You need your system to be involutive. If it isn't, then the system has to prolonged (differentiated to produce a higher order system); the Cartan-Kuranishi theorem states that a finite number of prolongations will produce either an involutive system, or an inconsistency.  This is explained in Seiler's book mentioned previously, which also includes worked examples. 
A: The Cartan--Kaehler theorem gives sufficient conditions for the existence of local solutions of any real analytic system of partial differential equations. However, the solutions are only local. See Bryant, Chern, Gardiner, Goldschmidt and Griffiths, Exterior Differential Systems, MSRI.
A: Because some of your operators are not linear, I don't have a definitive answer. But Haim Brézis considered problems of the form
$$\max(A_1u-f_1,A_2u-f_2)=0$$
where $A_1,A_2$ are second-order elliptic operators. See his paper The Hamilton-Jacobi-Bellman equations and variational inequalities. Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), pp. 385–395, Pitagora, Bologna, 1979. See also
(Brézis + L. C. Evans) in Arch. Rat. Mech. Anal. 70 (1979), no. 1, 1–13,
P.-L. Lions. Some problems related to the Bellman-Dirichlet equation for two operators.
Comm. Partial Differential Equations 5 (1980), no. 7, 753–771. 
