I'm reading Evans textbook, on chapter 8 there is a problem about multivalued PDE page 523 problem 8.7.14:
Show the variational inequality
$$\int_U Du\cdot D(w-u) \, dx\geq \int_U f(w-u) \, dx$$
For all $w\in H_0^1(U)$, $w\geq h$ a.e. in $U$. Can be rewritten as
$$-\Delta u+\beta (u-h)\ni f$$
For the multivalued function
\begin{equation} \beta(x) = \begin{cases} 0, & \text{if } x>0, \\ (-\infty,0], & \text{if } x=0, \\ \emptyset, & \text{if } x<0. \end{cases} \end{equation}
Who can explain this formula for me? Or give some reference about this. Thank all of you!