I found the following the following differential equation in the context of a Game Theory problem. I was wondering if this is related to any known family of equations or whether there is any hint about properties it might have. I am looking for functions $f:[0,1]^2 \rightarrow [0,1]^2$ satisfying:


$x\_1 \frac{\partial f\_1}{\partial x\_1} + x\_1 \frac{\partial f\_2}{\partial x\_1} = \frac{\partial f\_2}{\partial x\_1}$


$x\_2 \frac{\partial f\_1}{\partial x\_2} + x\_2 \frac{\partial f\_2}{\partial x\_2} = \frac{\partial f\_1}{\partial x\_2}$



Structure of solutions of a PDE from a game theory problem
Apply $\partial\_2$ to the first, $\partial\_1$ to the second and sum. You find
$$(x\_1+x\_2-1)\partial\_1\partial\_2(f\_1+f\_2)=0.$$
Away from the line $L:\,x\_1+x\_2=1$, you have $\partial\_1\partial\_2(f\_1+f\_2)=0$. Insert this in the original equations and you obtain $\partial\_1\partial\_2f\_j=0$. Whence
$$f\_i(x)=a\_{i1}(x\_1)+a\_{i2}(x\_2).$$
The functions $a\_{ij}$ obey to
$$x\_1(a\_{11}'+a\_{21}')=a\_{21}',\qquad x\_2(a\_{12}'+a\_{22}')=a\_{12}'.$$
This tells us that there exist potentials $p\_j(x\_j)$ such that for instance
$$a\_{11}=p\_1''-x\_1p\_1'+p\_1,\quad a\_{21}=x\_1p\_1'-p\_1.$$


If you are interested in what happens across the line $L$, you have to write jump relations (Rankine--Hugoniot conditions). For instance
$$x\_1\sum\_{i,j}[a\_{ij}]=[a\_{21}+a\_{22}],\qquad x\in L.$$


