Question

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:

$x_1 \frac{\partial f_1}{\partial x_1} + x_1 \frac{\partial f_2}{\partial x_1} = \frac{\partial f_2}{\partial x_1}$

$x_2 \frac{\partial f_1}{\partial x_2} + x_2 \frac{\partial f_2}{\partial x_2} = \frac{\partial f_1}{\partial x_2}$

Answer 1

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 $$x_1\sum_{i,j}[a_{ij}]=[a_{21}+a_{22}],\qquad x\in L.$$

Answer 2

For functions $g_1(x_1,x_2)=f_1(x_1^2,x_2^2-2x_2)$, $g_2(x_1,x_2)=f_2(x_1^2-2x_1,x_2^2)$ the system is linear:

$\partial_{x_1}g_1+ \partial_{x_1}g_2=0$,

$\partial_{x_2}g_1+ \partial_{x_2}g_2=0$.