# A boundary condition

Mathematics Asked by Lorenzo Andreaus on December 15, 2020

Given $$dcolon [0,1]to mathbb{R}^2$$, with $$d(0)=d(1)=0$$, I want to find a function $$ccolon [0,1]^2to mathbb{R}^2$$ such that $$c(x,0)-c(1,0)+c(0,1)-c(x,1)+c(1,x)-c(0,x)=d(x)$$, for all $$xin [0,1]$$.
I was thinking about consider $$c(x,0)-c(1,0)$$, $$c(0,1)-c(x,1)$$ and $$c(1,x)-c(0,x)$$ as the integral of $$c$$ along some paths in $$mathbb{R}^2$$, and think of $$c$$ as some kind of a potential function, but I don’t know how to proceed.

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