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.
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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. |
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Also note that the data define a Lipschitz function on the boundary of the square, and recall that any real-valued Lipschitz function defined on a subset of a metric space always admits a Lipschitz extension to the whole metric space, with the same Lipschitz constant. There exists the minimal and the maximal such extension (here's the maximal one). |
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