Consider the difference $$\frac{((u(t,x)-g(x))^+)^{1/k}}{\lambda} - \frac{((w(t,x)-g(x))^+)^{1/k}}{\lambda},$$
where $\lambda >0$ (small), $k \in \mathbb{N}$, $u,w$ are bounded and continuous functions on $[0,T] \times \mathbb{R}^N$, $g$ is a bounded and continuous function on $\mathbb{R}^N$, and $(\cdot)^+$ is the positive part function.
Questions:
- Is $\frac{((u(t,x)-g(x))^+)^{1/k}}{\lambda}$ lipschitz and convex?
- Can we have an estimate like $$\frac{((u(t,x)-g(x))^+)^{1/k}}{\lambda} - \frac{((w(t,x)-g(x))^+)^{1/k}}{\lambda} \le C||u(t,x)-w(t,x)||_{L^\infty}.$$ If not, what kind of estimates can we have?
- In points 1. and 2., do we have the same results if we consider $(\cdot)^-$ instead of $(\cdot)^+$ (wher $(\cdot)^-$ is the negative part function)?