# The jump set of $SBV$ function for different value of parameter in image denoising problem

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!

• I would look at "Singular Sets of Minimizers for the Mumford-Shah Functional" by Guy David. – Dirk Jun 27 '16 at 20:59
• @Dirk I know this book sir. I think it just discussd the properties of $S_{u_\alpha}$ a lot, but it does not discuss the relation between $S_{u_0}$ and $S_{u_\alpha}$... – JumpJump Jun 27 '16 at 21:05
• Ah, OK. It's been some time since I had this book in my hands. – Dirk Jun 27 '16 at 21:41
• You can also take a look at this paper: tuomov.iki.fi/mathematics/jumpset.pdf – Christian Clason Jun 28 '16 at 7:04
• @ChristianClason Thank you sir. This paper discusss my problem $(1)$ in $L^1$ setting, i.e., for ROF not MS, but it very useful. Thank you! Btw, do you have an idea that is there some paper disscuss about $(2)$? – JumpJump Jul 5 '16 at 14:30