So here is my problem,
Let $D:=[-d,d]\times[-d,d]$ and $C_0^{\infty}$(D) be the set of all smooth functions with compact support in $D$ which are zero on the boundary of $D$.
Moreover we have the two following inner products.
$$\langle\cdot,\cdot\rangle_0:L^2(D)\times L^2(D)\rightarrow\mathbb R$$ $$(f,g)\mapsto\int_Dfgdx$$ and $$\langle\cdot,\cdot\rangle_1:C^{\infty}_0(D)\times C^{\infty}_0(D)\rightarrow\mathbb R$$ $$(f,g)\mapsto\int_D\partial _{x_1}f\partial _{x_1}g+\partial _{x_2}f\partial _{x_2}gdx$$
Now I want to show the following inequality,
$$||f||_0\leq2d||f||_1$$ for all $f\in C^{\infty}_0(D)$.
This was my attempt,
For some $f\in C^{\infty}_0(D)$ we have,
$$4d^2||f||_1^2=||1||_0^2||f||_1^2=||1||_0^2(||\partial_{x_1}f||_0^2+||\partial_{x_2}f||_0^2)=||1||_0^2||\partial_{x_1}f||_0^2+||1||_0^2||\partial_{x_2}f||_0^2$$ By applying Cauchy-Schwarz it follows,
$$||1||_0^2||\partial_{x_1}f||_0^2+||1||_0^2||\partial_{x_2}f||_0^2\geq\langle1,\partial_{x_1}f\rangle_0^2+\langle1,\partial_{x_2}f\rangle_0^2=(\int_D\partial_{x_1}fdx)^2+(\int_D\partial_{x_2}fdx)^2$$
Now I know I could use the following identity to get $||f||_0^2$ somewhen.
$$f(x_1,x_2)=\int_{-d}^{x_1}\partial_{x_1}f(s,x_2)ds$$
But everytime I am applying the equality I am ending up having too many integrals in the end...
Could someone help me? Thanks!