I need some help with this exercise about distributional derivatives:
If we have $N=2$, and a function $g=\chi_{C}$, where $\chi$ is the characteristic function, and $C$ is the unitary cube $C=\lbrace{(x,y): \max\lbrace{|x|,|y|\rbrace}\leq 1\rbrace}$,
Compute the distributional derivatives $D^{(1,1)}g$ and $D^{(1,0)}g$.
Thanks a lot for any help.
I arrived at the following results, but i'm not sure if they are ok :
$\bullet$For $D^{(1,0)}g$: $<T_g^{(1,0)},\varphi>=-<T_g,\varphi^{(1,0)}>=<\chi_I,\varphi>$
where $I$ is the set $\lbrace{(\pm 1,y): y\in[-1,1]\rbrace}$.
$\bullet$ For $D^{(1,1)}g$: $<T_g^{(1,1)},\varphi>=+<T_g,\varphi^{(1,1)}>=<\chi_I\cdot\chi_S,\varphi>$
where $S=\lbrace{((x,\pm1):x\in[[-1,1]\rbrace}$
Best Answer
For the sake of completeness and to correct the question, here are the answers:
$$ (D^{(1,0)}T_g)(\varphi) = -\int_{-1}^1 \varphi(-1,y)-\varphi(-1,y) dy \,\text{ and} $$ $$ D^{(1,1)}T_g = \delta_{(1,1)} - \delta_{(-1,1)}- \delta_{(1,-1)} +\delta_{(-1,-1)}. $$