Let $U \subset \mathbb R^2$ be compact and consider
- $h:U \times (0,T) \to \mathbb R$ a uniformly bounded function such that: $0 \lt c_1 \le h \le 1-c_2 $
- $f_1,f_2 \in W^{2,1}_p(U \times (0,T))$ for each $p \ge 1$
I'm trying to understand why:
$\int_{\{x:f_1(x,\cdot) \gt 0\}} (1-h)dy -\int_{\{x:f_2(x,\cdot) \gt 0\}} (1-h)dy \le Cd_H(\{x:f_1(x,\cdot) \gt 0\},\{x:f_2(x,\cdot) \gt 0\})(*)$
where $d_H(\{x:f_1(x,\cdot) \gt 0\},\{x:f_2(x,\cdot) \gt 0\})$ denotes the Hausdorff distance
As I see it,
$\int_{\{x:f_1(x,\cdot) \gt 0\}} (1-h)dy -\int_{\{x:f_2(x,\cdot) \gt 0\}} (1-h)dy \le \\ (1-c_1)\vert \{x:f_1(x,\cdot) \gt 0\} \vert -c_2\vert \{x:f_2(x,\cdot) \gt 0\} \vert (**)$
QUESTIONS:
- How is $(*)$ related to $(**)$?
- Why can the Hausdorff distance be defined for the set $\{x:f_i(x,\cdot) \gt 0\}$ since it's not compact?
I'm afraid that my questions are quite trivial and silly but I'm not familiar at all to the Hausdorff distance and its properties. So, I would appreciate any help as well as a good reference in order to study more about this topic.
Thanks a lot in advance