I have a problem with the local analysis of two-variable functions which have absolute value. For example, how can I study nature of point $(0,0)$ in $f:\mathbb{R}^2 \to \mathbb{R}, f(x,y)=sin(xy+|xy|)$? Or in the function $f:\mathbb{R}^2 \to \mathbb{R}, f(x,y)=e^{sin(|xy|)}$?
I think that in both cases $(0,0)$ is a local minimum, but I don't know how to prove it. Typically, I use the Hessian matrix to do the analysis, however in this case I don't think it will help me...
This answer is example-specific. I don't think that there is one good method for investigating the extrema of non-differentiable functions.
For the first example note that $$xy + |xy| = \begin{cases}2xy,\quad &xy> 0,\\0,\quad &xy \leq 0.\end{cases}$$ In particular, the argument of the sine function is always non-negative, therefore $\sin(xy + |xy|)$ has a local minimum at $(0,0)$ because on the neighborhood of the origin $xy$ cannot be too large (and we will not obtain e.g. $\sin \frac 32\pi$). The minimum will not be strict though, as the function vanishes at all points $(x,y)$ for which $xy\leq 0$.
The other example is quite similar. The argument in the sine function is positive and small near the origin and thus the exponent $\sin |xy|$ will attain only non-negative values around $(0,0)$. Once again the minimum will not be strict.