The classical Mumford-Shah image denoisng problem study the minimizer of the following functional, for each $\alpha>0$ where $\Omega\subset \mathbb R^2$ is open bounded with sommth boundary, $$ u_\alpha:=\operatorname{argmin}_{u\in SBV(\Omega)}\left\{\int_\Omega\left|{u-u_0}\right|^2dx+\alpha\left(\int_\Omega\left|{\nabla u}\right|^2dx+\mathcal H^1(S_u)\right)\right\} $$ where $u_0\in SBV(\Omega)$ and $\nabla u$ denotes the absolute continuous part of $Du$ which is the weak derivative of $u$ in the sense of Radon measure. $S_u$ denote the jump set of $u$.

Clearly, for $\alpha=0$, $u_0=u_0$, and for $\alpha=+\infty$, $S_\infty$ would be empty.

My question: Would it be possible to prove that, if $\alpha_1<\alpha_2$,we have $$ \mathcal H^1(S_{u_{\alpha_2}}\setminus S_{u_{\alpha_1}})=0 \tag 1 $$ hold?

Moreover, do we even know that, for any $\alpha>0$, $$ \mathcal H^1(S_{u_{\alpha}}\setminus S_{u_{0}})=0 \tag 2 $$ hold?

Please advise. If you know some past study about a similar problem, please direct me there.

Thank you!