I have a question about a proof in "Partial Differential Equation by Lawrence C. Evans". We look at the problem
$(1)\mbox{ }u_t+H(Du,x) = 0 \mbox{ in }\mathbb{R}^n\times (0,T] \mbox{ }$ and $u=g$ on $\mathbb{R}^n\times \{t=0\}$. On page 546, there is a Lemma called "Extrema at a terminal time", i.e.
Assume $u$ is a viscosity solution of $(1)$ and $u-v$ has a local max at a point $(x_0,t_0)\in \mathbb{R}^n\times (0,T]$. Then $v_t(x_0,t_0)+H(Dv(x_0,t_0),x_0)\le 0 (\ge 0)$
So the point is, allowing $t_0=T$.
In the proof, we assume $u-v$ has a local max at $(x_0,T)$. W.l.o.g this is a strict max. Now he defines a new function $\tilde{v}(x,t):=v(x,t)+\frac{\epsilon}{T-t}$ for $x\in\mathbb{R}^n$ and $0
Two questions:
- Why does the point $(x_\epsilon,t_\epsilon)$ exists and is convergent to $(x_0,T)$?
If we go one step further and define a new equation
$(2)\mbox{ }u+u_t+H(Du,x) = 0 \mbox{ in }\mathbb{R}^n\times (0,T] \mbox{ }$
Let the boundary conditions be nice, I think it is not important here. Is there a similar Lemma to the one above and if so, how do we have to choose $\tilde{v}$? The argument, I guess, is then the same.
Thanks
math