I post this on MSE too. Sorry for this short cross posting. But I stuck in this problem for a while... I don't know what the right tool should be used to handle the moving domain problem...
Let $I:=(0,1)$ be given. Let $\bar u\in C^\infty(\bar I)$ be given. Define the energy functional, where $x\in I$ is fixed, $$ T(u,x):=\int_{I\setminus \{x\}}|u'|^2dx+\int_I|u-\bar u|^2dx,\text{ for }u\in H^1(I\setminus \{x\}) $$ and the function $t$: $I\to\mathbb R$ by $$ t(x):=\inf\{ T(u,x):\,\,u\in H^1(I\setminus \{x\}\} $$ Hence, $t(x)$ is well defined.
My question: do we have $t(x)$ is continuous? If yes, what the minimum assumption we need to put on $\bar u$. So far I give $\bar u\in C^\infty(\bar I)$, but it would be ideal that $\bar u\in BV(I)$.
Moreover, I am also wondering that if I define $$ u_x:=\operatorname{argmin}\{ T(u,x):\,\,u\in H^1(I\setminus \{x\}\} $$ Then do I have $u_x$ is continuous in $L^1$ sense? i.e., if $x\to x_0$, then $u_x\to u_{x_0}$ in $L^1$.
Thank you!
