Let $r: \mathbb{R} \rightarrow \mathbb{R}$ be a square integrable differentiable real function with square integrable derivative and let $\alpha = \int_{-\infty}^{+\infty}[r'(x)]^2dx$, where $r'$ denotes the derivative of r.
Now let $x = \sum_{i=1}^N (y_i-k)^2$, for $k, y_i \in \mathbb{R}, i \in \mathbb{Z}$ and define $u: \mathbb{R}^N \rightarrow \mathbb{R}$ such that $u(\vec y) = r(x)$, for all $\vec y = (y_1, y_2, ..., y_N)$. I want to compute the following integral as a function of $\alpha$ and $k$:
$\int_{\vec y \in \mathbb{R}^N}\left|\frac{d}{d \vec y}u(\vec y)\right|^2d \vec y \tag{1}$
Note: $\frac{d}{d \vec y}$ denotes the gradient of $r(\vec y)$.
Could anyone give me a help?
PS: some answers for the comments below:
1) Do you have any specific $r$ in mind? Actually no, but it would be of great help if one could solve for $r(x) = e^{-c^2 x}$.
2) Why $i$ goes from $1$ to $N$? It's just a choice. It could be $i=0$ to $N-1$. The important is that there are finite values for $i$.
3) What if $N = 1$? If $N=1$, then $x = y_1$.
4) Where it all comes from? This is part of a multidimensional cost function I need to minimize by varying the value of $k$. Ideally, it would be very great if I could write the cost function in terms of $k$ and $\alpha$ because of computational performance. Actually, I need to minimize the euclidean norm of the gradient function of $u$ by varying the unconstrained parameter $k$.
Some work on this until now:
$|\frac{d}{d \vec y} u(\vec y)|^2 = \sum_{j=1}^N [\frac{d}{d y_j}u(\sum_{i=1}^N (y_i-k)^2)]^2\tag{2}$.
So, the integral can be rewritten as:
$\sum_{j=1}^N \int_{-\infty}^{\infty} [\frac{d}{d y_j}u(\sum_{i=1}^N (y_i-k)^2)]^2 d y_j\tag{3}$
Applying the derivative:
$\frac{d}{d y_j}u(\sum_{i=1}^N (y_i-k)^2)=2(y_j-k) r'(\sum_{i=1}^N (y_i-k)^2)\tag{4}$
Replacing again in the integral, we get:
$\sum_{j=1}^N \int_{-\infty}^{\infty} [\frac{d}{d y_j}u(\sum_{i=1}^N (y_i-k)^2)]^2 d y_j = \sum_{j=1}^N \int_{-\infty}^{\infty} [2(y_j-k) r'(\sum_{i=1}^N (y_i-k)^2)]^2 d y_j \tag{5}$
From now, I would make a change of variables $x = \sum_{i=1}^N (y_i-k)^2$