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I am considering a problem of a two-dimensional ODE involving Karush, Kuhn and Tucker conditions on one of the unknowns. After a few algebraic manipulations, I end up having to solve the following PDEs: $$ \begin{aligned}&Q(x,y)=\frac{\partial P(x,y)}{\partial x}y+\frac{\partial P(x,y)}{\partial y}(f(x,y)+\lambda)\\ &f(x,y)=\frac{\partial Q(x,y)}{\partial x}y+\frac{\partial Q(x,y)}{\partial y}(g(x,y)+\lambda)\end{aligned}$$ with: $$ x-d\leq 0 \quad;\quad \lambda\geq 0 \quad;\quad (x-d)\lambda=0$$ In these equations, $d$, $f(x,y)$, and $g(x,y)$ are known; $P(x,y)$, $Q(x,y)$, and $\lambda$ should be found. I am not detailing the boundary conditions. Have you ever seen PDEs of this type ($\lambda$ is problematic)? If yes, are you aware of a numerical strategy that would solve them?

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