Is it possible to define explicitly a Lipschitz function $f:[a,b]\times[c,d]\rightarrow \mathbb{R}$ in term of $f(a,\cdot)$, $f(b,\cdot)$, $f(\cdot,c)$, $f(\cdot,d)$ if I know these functions and they are Lipschitz? 
Thanks.



Lipschitz function
If $a=c=0$ and $b=d=1$, define $f$ by 
affine interpolation between $f(x+y,0)$ and $f(0,x+y)$ if $x+y\leq 1$ respectively
$f(x+y-1,1)$ and $f(1,x+y-1)$ if $x+y\geq 1$.


The general case can be reduced to the previous one by an affine transformation.


