# How to minimise objective function with H1 constraint?

I have a cost function that I try to minimise:

$$\Pi \sim ||y - Ax||^2 + ||\nabla x || ^2$$

The gradient is there to constraint that my solution at minimum $x_{min}$ is continuous in 1st derivative.

Now dimension-wise, $y$ is a vector with $m \times 1$, $A$ is a matrix mapping x to y-space with dimension $m \times n$ and $x$ is a vector of $n \times 1$.

Ideally, I am trying to make this optimise problem into a matrix form of:

$$Kx = b$$

and then hopefully can apply CG on it. But I fail at deriving the matrix $K$ itself.

Any suggestions?