So, let $\Omega\subset \mathbb R^2$ bounded and consider a sequence of functions $\{u_k\}_{k\in\mathbb N}\subset BV(\Omega)$ and $u\in BV(\Omega)$ such that $u_k\rightarrow u$ weakly* in $BV(\Omega)$. Let $S_u,S_{u_k}$ be the approximate discontinuity sets of $u,u_k$ respectively and fix $x_0\in \Omega_1:=\Omega \setminus \left(S_u\cup S_{u_1}\cup\cdots\right)$. The approximate limits $\tilde u$ and $\tilde u_k$ are well-defined in $\Omega_1$ (I am following the notation of Ambrosio-Fusco-Pallara). Does it hold that $\tilde u_k(x_0)\rightarrow \tilde u(x_0)$, possibly up to a subsequence?

I think that Egorov's theorem has to come up at some point but I am not able to include $x_0$ to the set where the functions converge uniformly.

Moreover, I am not sure if it is of any help, but we may assume that $u,u_k$ are pointwise uniformly bounded (ex. they have values in $[0,1]$).

Any help is warmly welcome!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.