Jtdylktuy

I am looking the solution of the exercice 7.4 in the book Analysis of Lieb--Loss:

Suppose that $fin H^1(mathbb R^n)$. Show that for each $1leq ileq n$
$$
int_{mathbb R^n}|partial_if|^2=lim_{tto 0}frac{1}{t^2}int_{mathbb R^n}|f(x+tmathbf{e}_i)-f(x)|^2{rm d}x
$$
where $mathbf{e}_i$ is the unit vector in the direction $i$.

My approach is to approximate $fin H^1(mathbb R^n)$ by a sequence of $C_c^infty(mathbb R^n)$ functions. Then we can check the above identity for this smooth with compact support function. But then I do not know how to go back to the function in $H^1(mathbb R^n)$.

You're right - since there is a limit $t to 0$ involved, it's not obvious how to apply the density of $C_c^infty$ in $W^{1,2}$. A priori it's not even clear that the limit exists. You can do it as follows.

To simplify the notation, let
$$
partial^t_i f(x) := frac{f(x + t e_i) - f(x)}{t}.
$$

Step 1. Check that $| partial_i^t f |_{L^2} le | partial_i f |_{L^2}$ for each $f in C_c^infty$ and $t > 0$.

Step 2. By density, the same holds also for $f in W^{1,2}$. This shows that
$$
| partial_i f |_{L^2} ge limsup_{t to 0} | partial_i^t f |_{L^2}.
$$

Step 3. Consider $f in W^{1,2}$. Assume that for some sequence $t_k to 0$, $partial_i^{t_k} f$ tends weakly in $L^2$ to some function $g in L^2$, then $g$ is the distributional partial derivative $partial_i f$. To see it, consider the integral $int partial_i^t f varphi$ with some test function $varphi in C_c^infty$, apply discrete integration by parts and take the limit.

Note that weak convergence gives you $| g |_{L^2} le liminf_{k to infty} | partial_i^{t_k} f |_{L^2}$ and so
$$
| partial_i f |_{L^2} le liminf_{t to 0} | partial_i^t f |_{L^2}.
$$