Explanation of a passage about a smooth approximation to $L^p$ function

Question

I'm reading J.L Vazquez "Porous Medium Equation" book. In it, he says the following:

We are given a function $a:\Omega \times (0,T) \to \mathbb{R}$ such that $a \geq 0$. We find a smooth approximation $a_\epsilon$ of $a$ such that $\epsilon \leq a_\epsilon \leq K$.

...

We now have to examine the way we construct the approximation. We do it like this: given $\epsilon > 0$, we select a height $K > \epsilon > 0$, and define $a_{K, \epsilon} = \min\{K, \max\{\epsilon, a\}\}$ (we will be taking $K$ very large and $\epsilon$ small), and then we take smooth approximations $a_n \to a_{K, \epsilon}$ in $L^p$.

Later on, the author chooses $n=n(\epsilon, K)$ large enough to achieve something.

Question: I'm a bit confused about what he uses for the approximation $a_\epsilon$. It seems when he explains how the approximation is done, he eventually ends up $a_n$ -- is this supposed to be relabelled $a_\epsilon$? And the height K depends on $\epsilon$.. can someone write this better? Mainly I am interested in: how does $K$ depend upon $\epsilon$?

Answer 1

No, $K$ does not depend on $\epsilon$. Rather, both $K$ and $\epsilon$ depend on the quality of approximation we want. A precise statement would be:

Given $\delta>0$, we can find two positive numbers $\epsilon, K$ and a smooth function $b$ such that $\epsilon\le b\le K$ and $\|a-b\|_{L^p}<\delta$.

We can't really control $\epsilon, K$; if the given $\delta$ is small, then $\epsilon$ will have to be very small, and $K$ very large.

How to prove the above: define $a_{K, \epsilon} = \min\{K, \max\{\epsilon, a\}\}$ as the author does. Observe that $$\|a_{K, \epsilon} -a \|_p^p = \int_{a<\epsilon} a^p + \int_{a>K } a^p$$

• Take $\epsilon$ small enough so that the first integral is less that $(\delta/3)^p$.
• Take $K$ large enough so that the second integral is less that $(\delta/3)^p$.
• Observe that $\|a_{K, \epsilon} -a \|_p <2\delta/3$.
• Convolve $a_{K,\epsilon}$ with a bump function (as in your other question) to obtain a smooth function $b$ such that $\epsilon \le b\le K$ and $\|a_{K, \epsilon} -b \|_p < \delta/3 $

This function $b$ is what we wanted.

The author probably had a good reason to call the approximating function $a_\epsilon$, although I don't see this reason from the quoted passage.