MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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:

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

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

share|cite|improve this question
up vote 5 down vote accepted

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

share|cite|improve this answer
Amazing! Thanks. – Renato Apr 21 '11 at 13:22

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:

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

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.