i have a limited domain $X \subset \mathbb{R}^{n}$, and two functions, $u$ continuous in $X$ and $ f \in C^{2}(X)$, such that $u-f$ has local minimum in a given $x_{0} \in X$, ie for some $p> 0$, $x_{0}$ minimizes $u-f$ in ball with center $x_{0}$ and radius $p$.
Aff: Given $y \in B_{r}(0)$ with $r> 0$ small enough, the function $f_{y}(x) = f (x + y)$ is such that $u-f_{y}$ has a local minimum at some point $x_{y} \in B_{p} (x_{0})$
ATTEMPT: Well, one of the first things to consider is that this $r> 0$ should be taken less than $2p$, according to the triangular inequality, so that ensures that $(x + y) \in B_{p}(x_{0})$. In addition, if $u-f$ are continuous there is at the infimum of $u-f_{y}$ on $\bar{B}_{p}(x_{0})$, if the point where the infimum is reached is inside the ball, ok. Now what would be the idea of argue over the possibility of the point where the infimum is reached is in the $\partial B_{p}(x_{0})$ I suspect you should use that $u-f$ has minimum local at $x_{0}$
Could anyone give a help?
NOTE: Geometrically, I am saying that if the graph of the $C^{2}$ function touches the graph below $u$, given a little "push" on the graph of the function $f$ in a given neighborhood, I still have u being touched underneath. by the graph of the function $C^{2}$.