Piecewise constant profiles minimizing total variation energies of Kobayashi-Warren-Carter type with fidelity

We consider a kind of total variation energy which measures jumps different from the conventional total variation energy. Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with $n\geq 1$. Let $u$ be in $BV(\Omega)$, i.e., its distributional derivative $Du$ is a finite Radon measure in $\Omega$ and let $|Du|$ denote its total variation measure. The total variation energy can be written of the form

$$TV(u)=\int_{\Omega\backslash J_{u}}|Du|+\int_{J_{u}}|u^{+}-u^{-}|\;d\mathcal{H}^{n-1},$$

where $J_{u}$ denotes the (approximate) jump set of $u$ and $u^{\pm}$ is a trace of $u$ from each side of $J_{u}$; $\mathcal{H}^{n-1}$ denotes the $n-1$ dimensional Hausdorff measure. For a precise meaning of this formula, see Section 2 and [AFP]. Let $K(\rho)$ be a strictly increasing continuous function for $\rho\geq 0$ with $K(0)=0$. We set

$$TV_{K}(u)=\int_{\Omega\backslash J_{u}}|Du|+\int_{J_{u}}K\left(|u^{+}-u^{-}|\right)d\mathcal{H}^{n-1}.$$

For a given function $g\in L^{2}(\Omega)$, we are interested in a minimizer of

$$TV_{Kg}(u)=TV_{K}(u)+\mathcal{F}(u),\quad\mathcal{F}(u)=\frac{\lambda}{2}\int_{\Omega}|u-g|^{2}\;dx,$$

where $\lambda>0$ is a constant. The term $\mathcal{F}$ is often called a fidelity term. If $TV_{K}=TV$, the functional $TV_{g}(u)=TV(u)+\mathcal{F}(u)$ is often called Rudin-Osher-Fatemi functional since it is proposed by [ROF] to denoise the original image whose grey-level values equal $g$. For $TV_{g}$, there always exists a unique minimizer since the problem is strictly convex and lower semicontinuous in $L^{2}(\Omega)$. We are interested in regularity of a minimizer of $TV_{Kg}$ assuming some regularity of $g$. This problem is well studied for $TV_{g}$ started by [CCN]. Let $u_{*}$ be the minimizer of $TV_{g}$ for $g\in BV(\Omega)$. In [CCN], it is shown that $J_{u_{*}}\subset J_{g}$ and $u_{*}^{+}(x)-u_{*}^{-}(x)\leq g^{+}(x)-g^{-}(x)$ for $\mathcal{H}^{n-1}$-a.e. $x\in J_{u}$. In particular, if $g$ has no jumps, so does $u_{*}$. This type of results is extended in various setting; see a reviewer paper [GKL].

In this paper, we shall show that a minimizer of $TV_{Kg}$ may have jumps even if $g$ has no jumps for some class of subadditive function $K$ including

$$K(\rho)=\frac{\rho}{1+\rho}$$

as a particular example when $\Omega$ is an interval. This type of $TV_{K}$ appears as a kind of singular limit of the Kobayashi-Warren-Carter energy [GOU], [GOSU]. Actually, we shall prove a stronger result saying that a minimizer is a piecewise constant function with finitely many jumps for $g\in C(\overline{\Omega})$ when $\Omega$ is a bounded interval. Here is a precise statement.

For a function measuring a jump, we assume that

1. (K1)

   $K:[0,\infty)\to[0,\infty)$ is a continuous, strictly increasing function with $K(0)=0$.

2. (K2)

   For $M>0$, there exists a positive constant $C_{M}$ such that

   $$K(\rho_{1})+K(\rho_{2})\geq K(\rho_{1}+\rho_{2})+C_{M}\rho_{1}\rho_{2}$$

   for all $\rho_{1},\rho_{2}\geq 0$ with $\rho_{1}+\rho_{2}\leq M$. In particular, $K$ is subadditive.

3. (K3)

   $\lim_{\rho\to 0}K(\rho)/\rho=1$.

If $K(\rho)=\rho$ so that $TV_{K}=TV$, $K$ satisfies (K1) and (K3) but does not satisfy (K2). If $K(\rho)=\rho/(1+\rho)$, $K$ satisfies (K2) as well as (K1) and (K3). Indeed, a direct calculation shows that

$$\frac{\rho_{1}}{1+\rho_{1}}+\frac{\rho_{2}}{1+\rho_{2}}=\frac{\rho+2\rho_{1}\rho_{2}}{1+\rho+\rho_{1}\rho_{2}}=\frac{\rho}{1+\rho}+\frac{(2+\rho)\rho_{1}\rho_{2}}{(1+\rho)(1+\rho+\rho_{1}\rho_{2})},\quad\rho=\rho_{1}+\rho_{2}.$$

Thus, (K2) follows.

We note that we do not assume that $g\in BV(a,b)$. If $g$ is non-decreasing or non-increasing, we have a sharper estimate for $m$.

To show Theorem 1.1 and also Theorem 1.2, we introduce the notion of a coincidence set $C$, which is formally defined by

$$C=\left\{x\in[a,b]\bigm{|}U(x)=g(x)\right\}.$$

It turns out that a minimizer $U$ of $TV_{Kg}$ is always continuous on $C$ and outside this set $C$, $U$ is piecewise constant and

$$\sup_{[a,b]}U\leq\sup_{[a,b]}g,\quad\inf_{[a,b]}U\geq\inf_{[a,b]}g.$$

(1.1)

Moreover, we are able to prove that $U$ has at most one jump on $(\alpha,\beta)$ if $(\alpha,\beta)\cap C=\emptyset$ and $\alpha,\beta\in C$. We also prove that if $\alpha,\beta\in C$ with $\alpha<\beta$ is too close, $U$ must be monotone on $(\alpha,\beta)$. Here, $(\alpha,\beta)$ may include some point of $C$. To show these properties, it suffices to assume the subadditivity of $K$ instead of (K2). It includes the case $TV_{K}$, where $K(\rho)=\rho$ so that $TV_{K}$ is the standard total variation.

We shall prove that if $\alpha\in C$ and $\beta\in C$ with $\alpha<\beta$ is too close, $U$ must be a constant on $(\alpha,\beta)$. A key step (Lemma 4.7) is to show that for a non-decreasing minimizer $U$ on $(a,b)$ and $\alpha,\beta\in C$ with $\alpha<\beta$, the estimate

$$\beta-\alpha>2C_{M}/\lambda$$

holds provided that there is $\gamma\in C\cap(\alpha,\beta)$ and $(\gamma,\beta)\cap C=\emptyset$ and that $\rho_{2}:=U(\beta)-U(\gamma)\geq\rho_{1}:=U(\gamma)-U(\alpha)$, where $C_{M}$ is a constant in (K2). Here is a rough strategy to prove the statement. Since $(\gamma,\beta)\cap C=\emptyset$, there is exactly one jump point $x_{1}$ in $(\gamma,\beta)$. We compare $TV_{Kg}(U)$ and $TV_{Kg}(v)$ in $(\alpha,\beta)$, where $v$ equals $U(\alpha)$ on $(\alpha,x_{1})$ and equals $U$ on $(x_{1},\beta)$. It is not difficult to prove

$$TV_{K}(U)-TV_{K}(v)\geq K(\rho_{1})+K(\rho_{2})-K(\rho_{1}+\rho_{2})\geq C_{M}\rho_{1}\rho_{2}$$

(1.2)

since $\rho_{1}+\rho_{2}\leq M$ by (1.1). The proof for

$$\frac{2}{\lambda}\left(\mathcal{F}(U)-\mathcal{F}(v)\right)\geq-\rho_{1}\rho_{2}(x_{1}-\alpha)$$

(1.3)

is more involved. It is not difficult to show

$$\int_{\gamma}^{x_{1}}\left\{(U-g)^{2}-(v-g)^{2}\right\}dx\geq-\rho_{1}(\rho_{1}+\rho_{2})(x_{1}-\gamma).$$

(1.4)

The proof for

$$\int_{\alpha}^{\gamma}\left\{(U-g)^{2}-(v-g)^{2}\right\}dx\geq-\rho_{1}^{2}(\gamma-\alpha)$$

(1.5)

is more difficult. If $g$ is non-decreasing, we are able to prove

$$g(x)-U\leq U(x_{2}+0)-U(x_{2}-0)\quad\text{for}\quad x\in F,$$

where $F=[x_{0},x_{2}]\subset[a,b)$ is a maximal closed interval (called a facet) such that $U$ is a constant in the interior $\operatorname{int}F$ of $F$. For general $F$, one only expects such an inequality just in the average sense, i.e.,

$$\int_{x_{0}}^{x_{2}}\left(g(x)-U\right)dx\leq\left(U(x_{2}+0)-U(x_{2}-0)\right)(x_{2}-x_{0}).$$

(1.6)

The estimate (1.5) follows from (1.6). Combining (1.4) and (1.5), we obtain (1.3) since $\rho_{2}\geq\rho_{1}$. The estimates (1.2) and (1.3) yield

$$TV_{Kg}(U)-TV_{Kg}(v)\geq\rho_{1}\rho_{2}\left(C_{M}-(x_{1}-\alpha)\lambda\bigm{/}2\right).$$

If $\beta-\alpha\leq 2C_{M}/\lambda$, $U$ cannot be a minimizer. Similarly, we are able to prove that if $U$ is continuous on $(\alpha,\beta)$ with $\alpha,\beta\in C$, then $U$ is a constant on $(\alpha,\beta)$. These observations yield Theorem 1.1. Theorem 1.2 can be proved similarly to Theorem 1.1 by noting that a minimizer $U$ is a monotone function.

We also give a quantative estimate for $TV_{Kg}$ for monotone data $g\in C[a,b]$. It is easy to estimate $TV_{Kg}(u)$ for a piecewise constant $u$ from below. We approximate a general $BV$ function $u$ by piecewise constant functions $u_{m}$ so that $TV_{Kg}(u_{m})\to TV_{Kg}(u)$. Using such an approximation result, we establish an estimate of $TV_{Kg}$ for a general $BV$ function. We notice that such an estimate gives another way to prove Theorem 1.2.

It is not difficult to prove that $TV_{Kg}$ is lower semicontinuous in the space of piecewise constant functions with at most $k$ jumps. Since the space is of finite dimension, its bounded closed set is compact. By Weierstrass’ theorem, $TV_{Kg}$ admits a minimizer $v$ among piecewise constant functions with at most $k$ jumps with $\inf g\leq v\leq\sup g$. Thus, Theorem 1.1 guarantees the existence of a minimizer in $BV(a,b)$. The existence of minimizer of $TV_{Kg}$ itself can be proved for general essential bounded measurable function $g$, i.e., $g\in L^{\infty}(\Omega)$ for general Lipschitz domain in $\mathbb{R}^{n}$ since it is known [AFP] that $TV_{K}$ is lower semicontinuous in a suitable topology. We shall discuss this point in Section 2. Note that instead of (K2) subadditivity for $K$ is enough to have the existence of a minimizer.

We believe Theorem 1.1 extends for general $g\in L^{\infty}(a,b)$. In fact, we have a weaker version.

This follows from above-mentioned approximation result and Theorem 1.1. Since the uniqueness of minimizer is not guaranteed in general, there might exist another minimizer which is not piecewise constant although it is unlikely.

As shown in [GOU], [GOSU], $TV_{Kg}$ is obtained as a singular limit ($\varepsilon\downarrow 0$ limit) of the Kobayashi-Warren-Carter type energy

	$\displaystyle E_{\mathrm{KWC}g}^{\varepsilon}(u,v)$	$\displaystyle:=E_{\mathrm{KWC}}^{\varepsilon}(u,v)+\mathcal{F}(u)$
	$\displaystyle E_{\mathrm{KWC}}^{\varepsilon}(u,v)$	$\displaystyle:=\int_{\Omega}sv^{2}|Du|+E_{\mathrm{sMM}}^{\varepsilon}(v),\quad s>0$
	$\displaystyle E_{\mathrm{sMM}}^{\varepsilon}(v)$	$\displaystyle:=\frac{\varepsilon}{2}\int_{\Omega}|\nabla v|^{2}\;dx+\frac{1}{2\varepsilon}\int_{\Omega}F(v)\;dx$

by minimizing order parameter $v$; here $F(v)$ is a single-well potential typically of the form $F(v)=(v-1)^{2}$ and $s$ is a positive parameter. In fact, in one-dimensional setting [GOU], the Gamma limit of $E_{\mathrm{sMM}}^{\varepsilon}(v)$ under the graph convergence formally equals

$$E_{\mathrm{sMM}}^{0}(\Xi)=\sum_{i=1}^{\infty}2\left(G(\xi_{i}^{-}+G(\xi_{i}^{+})\right),\quad G(s)=\left|\int_{1}^{s}\sqrt{F(\xi)}\;d\xi\right|$$

if the limit of $v$ in $\Omega=(a,b)$ equals a set-valued function $\Xi$ of the form

$$\Xi(x)=\left\{\begin{array}[]{ll}&1,\quad x\notin\Sigma,\\ &[\xi_{i}^{-},\xi_{i}^{+}](\ni 1)\quad\text{for}\quad x\in\Sigma,\end{array}\right.$$

where $\Sigma$ is some (at most) countable set. The Gamma limit of $E_{\mathrm{KWC}}^{\varepsilon}$ equals

$$E_{\mathrm{KWC}}^{0}(u,\Xi)=\sum_{i=1}^{\infty}\left(s(\xi_{i,+}^{-})^{2}\left|u^{+}(x_{i})-u^{-}(x_{i})\right|+2\left(G(\xi_{i}^{-})+G(\xi_{i}^{+})\right)\right)+\int_{\Omega\backslash J_{u}}|Du|,$$

where $\xi_{+}=\max(\xi,0)$ and $x_{i}\in\Sigma$. Here for $u$, $L^{1}$ type limit is considered. If we minimize $E_{\mathrm{KWC}}^{0}(u,\Xi)$ by fixing $u$, $\xi^{+}$ must be one since $[\xi^{-},\xi^{+}]\ni 1$ and $G(1)=0$. Thus

	$\displaystyle\inf_{\Xi}E_{\mathrm{KWC}}^{0}(u,\Xi)$	$\displaystyle=\sum_{i=1}^{\infty}\min_{\xi>0}\left(s(\xi_{+})^{2}\left|u^{+}(x_{i})-u^{-}(x_{i})\right|+2G(\xi)\right)+\int_{\Omega\backslash J_{u}}|Du|$
		$\displaystyle=TV_{K}(u)+\int_{\Omega\backslash J_{u}}|Du|,$

where

$$K(\rho)=\min_{\xi}\left(s(\xi_{+})^{2}\rho+2G(\xi)\right).$$

(1.7)

In the case, $s=1$ and $F(v)=(v-1)^{2}$, a direct calculation shows that

$$K(\rho)=\min_{\xi>0}\left(\xi^{2}\rho+(\xi-1)^{2}\right)=\frac{\rho}{\rho+1}.$$

We shall prove that $K$ defined in (1.7) satisfies (K2) provided that

$$\varlimsup_{v\uparrow 1}F^{\prime}(v)\bigm{/}(v-1)<\infty$$

for $F$ if $F\in C(\mathbb{R})$ is non-negative and $F(v)=0$ if and only if $v=0$.

If we replace $\int sv^{2}|Du|$ by $\int sv^{2}|Du|^{2}$, the energy corresponding to $E_{\mathrm{KWC}g}^{\varepsilon}$ is nothing but what is called the Ambrosio-Tortorelli energy [AT]. Its singular limit is a Mumford-Shah functional

$$E_{\mathrm{MS}}(u,K):=s\int_{\Omega\backslash K}|\nabla u|^{2}+\mathcal{H}^{n-1}(K)+\mathcal{F}(u),$$

where $K$ is a closed set in $\Omega$ [AT], [AT2], [FL]. The existence of a minimizer is obtained in [GCL] by using the space of special functions with bounded variation, i.e., $SBV$ functions which is a subspace of $BV(\Omega)$.

A modified total variation energy $TV_{K}$ is not limited to a singular limit of the Kobayashi-Warren-Carter energy. In fact, $TV_{K}(u)$ like energy is derived as the surface tension of grain boundaries in polycrystals by [LL], where $u$ is taken as a piecewise constant (vector-valued) function; see also [GaFSp] for more recent development. The function $K$ measuring jumps may not be isotropic but still concave. In [ELM], $TV_{K}$ type energy for a piecewise constant function is also considered to study motion of a grain boundary. However, in their analysis, the convexity of $K$ is assumed.

This paper is organized as follows. In Section 2, we give a rigorous formulation of $TV_{K}$ and prove the existence of a minimizer of $TV_{Kg}$ for $g\in L^{\infty}(\Omega)$ for a general Lipschitz domain in $\mathbb{R}^{n}$. In Section 3, we study a profile of a minimizer $U$ for $TV_{Kg}$ including $TV_{g}$ outside the coincidence set. We also prove that $U$ is monotone in $(\alpha,\beta)$ with $\alpha,\beta\in C$ provided that $\alpha$ and $\beta$ is close. In Section 4, we prove that $\alpha,\beta\in C$ cannot be too close if $K$ satisfies (K2). We prove Theorem 1.1, Theorem 1.2 and Theorem 1.3. We also establish an estimate for $TV_{Kg}$ for monotone $g$. In Section 5, we shall discuss a sufficient condition that $K$ in (1.7) satisfies (K2).

In this section, after giving a precise definition of $TV_{K}$, we give an existence result for its Rudin-Osher-Fatemi type energy. The proof is based on a standard compactness result for $TV$ and a classical lower semicontinuity result for $TV_{K}$.

We recall a standard notation as in [AFP]. Let $\Omega$ be a domain in $\mathbb{R}^{n}$. For a locally integrable (real-valued) function $u$, we consider its total variation $TV(u)$ in $\Omega$, i.e.,

$$TV(u)=\sup\left\{\int_{\Omega}-u\operatorname{div}\varphi\;dx\biggm{|}\varphi\in C_{c}^{\infty}(\Omega,\mathbb{R}^{n}),\ \left|\varphi(x)\right|\leq 1\ \text{in}\ \Omega\right\},$$

where $C_{c}^{\infty}(\Omega,\mathbb{R}^{n})$ denotes the space of all $\mathbb{R}^{n}$-valued smooth functions with compact support in $\Omega$. An integrable function $u$, i.e., $u\in L^{1}(\Omega)$, is called a function of bounded variation if $TV(u)<\infty$. The space of all such function is denoted by $BV(\Omega)$, i.e.,

$$BV(\Omega)=\left\{u\in L^{1}(\Omega)\bigm{|}TV(u)<\infty\right\}.$$

By Riesz’s representation theory, one easily observe that $TV(u)$ is finite if and only if the distributional derivative $Du$ of $u$ is a finite Radon measure on $\Omega$ and its total variation $|Du|(\Omega)$ in $\Omega$ equals $TV(u)$.

We next define a jump discontinuity of a locally integrable function. Let $B_{r}(x)$ denote an open ball of radius $r$ centered at $x$ in $\mathbb{R}^{n}$. In other words,

$$B_{r}(x)=\left\{y\in\mathbb{R}^{n}\bigm{|}|y-x|<r\right\}.$$

For a unit vector $\nu\in\mathbb{R}^{n}$, we define a half ball of the form

$$B_{r}^{\pm}(x,\nu)=\left\{y\in B_{r}(x)\bigm{|}\pm\nu\cdot(y-x)\geq 0\right\},$$

where $a\cdot b$ for $a,b\in\mathbb{R}^{n}$ denotes the standard inner product in $\mathbb{R}^{n}$. Let $w$ be a locally integrable function in $\Omega$. We say that a point $x\in\Omega$ is a (approximate) jump point of $w$ if there exists a unit vector $\nu_{w}\in\mathbb{R}^{n}$, $w^{\pm}\in\mathbb{R}$, $w^{+}\neq w^{-}$, such that

$$\lim_{r\downarrow 0}\frac{1}{\mathcal{L}^{n}\left(B_{r}^{\pm}(x,\nu_{w})\right)}\int_{B_{r}^{\pm}(x,\nu_{w})}\left|w(y)-w^{\pm}\right|dy=0.$$

Here $\mathcal{L}^{n}$ denotes the Lebesgue measure in $\mathbb{R}^{n}$ so this integral is the average of $\left|w(y)-w^{\pm}\right|$ over $B_{r}^{\pm}(x,\nu_{w})$. The set of all jump points of $w$ is denoted by $J_{w}$ and called the (approximate) jump (set) of $w$. By definition, $J_{w}$ is contained in the set $S_{w}$ of (approximate) discontinuity point of $w$, i.e.,

$$S_{w}=\Biggl{\{}x\in\Omega\biggm{|}\lim_{r\downarrow 0}\frac{1}{\mathcal{L}^{n}\left(B_{r}(x)\right)}\int_{B_{r}(x)}\left|w(y)-z\right|dt=0\ \text{does not hold}\\ \text{for any choice of}\ z\in\mathbb{R}\Biggr{\}}.$$

By the Federer-Vol’pert theorem [AFP, Theorem 3.78], we know $\mathcal{H}^{n-1}$
$(S_{u}\backslash J_{u})=0$, where $\mathcal{H}^{m}$ denotes $m$-dimensional Hausdorff measure provided that $u\in BV(\Omega)$. Moreover, $S_{u}$ is countably $\mathcal{H}^{n-1}$-rectifiable, i.e., $S_{u}$ can be covered by a countable union of Lipschitz graphs up to an $\mathcal{H}^{n-1}$ measure zero set. In particular, $J_{u}$ is also countably $\mathcal{H}^{n-1}$-rectifiable. Quite recently, it is proved that $J_{u}$ is always countably $\mathcal{H}^{n-1}$-rectifiable if we only assume that $w$ is locally integrable [DN]. For $u\in BV(\Omega)$, the value $u^{\pm}$ can be viewed as a trace of $u$ on a countably $\mathcal{H}^{n-1}$-rectifiable set [AFP, Theorem 3.77, Remark 3.79] except $\mathcal{H}^{n-1}$ negligible set (up to permutation of $u^{+}$ and $u^{-}$). We now recall a unique decomposition of the Radon measure $Du$ for $u\in BV(\Omega)$ of the form

$$Du=D^{a}u+D^{s}u,\quad D^{s}u=D^{c}u+(u^{+}-u^{-})\cdot\nu_{u}\mathcal{H}^{n-1}\lfloor J_{u};$$

see [AFP, Section 3.8]. The term $D^{a}u$ denotes the absolutely continuous part and $D^{s}u$ denotes the singular part with respect to the Lebesgue measure. The term $D^{a}u=\nabla u\mathcal{L}^{n}$, where $\nabla u\in\left(L^{1}(\Omega)\right)^{n}$. The singular part is decomposed into two parts; $D^{c}u$ vanishes on sets of finite $\mathcal{H}^{n-1}$ measure. For a measure $\mu$ on $\Omega$ and a set $A\subset\Omega$, the associate measure $\mu\lfloor A$ is defined as

$$(\mu\lfloor A)(W)=\mu(A\cap W),\quad W\subset\Omega.$$

We now consider a total variation type energy measuring jumps in a different way. For $u\in BV(\Omega)$, we set

$$TV_{K}(u):=\left(TV_{K}(u,\Omega):=\right)\int_{\Omega\backslash J_{u}}|Du|+\int_{J_{u}}K\left(|u^{+}-u^{-}|\right)d\mathcal{H}^{n-1}.$$

Here the density function is assumed to satisfy following conditions.

1. (K1w)

   $K:(0,\infty)\to[0,\infty)$ is non-decreasing, lower semicontinuous.

2. (K2w)

   $K$ is subadditive, i.e., $K(\rho_{1}+\rho_{2})\leq K(\rho_{1})+K(\rho_{2})$.

3. (K3)

   $\lim_{\rho\to 0}K(\rho)/\rho=1$.

We consider the Rudin-Osher-Fatemi type energy for $TV_{K}$, i.e.,

$$TV_{Kg}(u):=TV_{K}(u)+\mathcal{F}(u),\quad\mathcal{F}(u)=\frac{\lambda}{2}\int_{\Omega}|u-g|^{2}\;dx$$

for $g\in L^{2}(\Omega)$.

In the rest of this section, we shall prove Theorem 2.1 by a simple direct method. We begin with compactness.

For a lower semicontinuity, we have

This is a special form of the lower semicontinuity result [AFP, Theorem 5.4]. We restate it for the reader’s convenience. We consider

$$F(u)=\int_{\Omega\backslash J_{u}}\varphi\left(|\nabla u|\right)dx+\beta|D^{c}u|(\Omega)+\int_{J_{u}}K\left(|u^{+}-u^{-}|\right)d\mathcal{H}^{n-1}.$$

See [AFP, Theorem 5.4]. Such a lower semicontinuity is originally due to Bouchitté and Buttazzo [BB]. In our setting $\varphi(t)=t$, $\beta=1$.

In this section, we discuss one-dimensional setting and study properties of coincidence set

$$C=\left\{x\in\Omega\bigm{|}U(x)=g(x)\right\}$$

of a minimizer $U$.

Let $\Omega$ be a bounded open interval, i.e., $\Omega=(a,b)$. We consider $TV_{Kg}(u)$ for $g\in C[a,b]$ for $u\in BV(a,b)$. Since $u$ can be written as a difference of two non-decreasing function, we may assume that $u$ has a representative that $J_{u}$ is at most a countable set and outside $J_{u}$, $u$ is continuous. Moreover, we may assume that $u$ is right (resp. left) continuous at $a$ (resp. at $b$). For $x\in J_{u}$, $u(x\pm 0)$ is well-defined.

For $K$, we assume

1. (K1)

   $K:[0,\infty)\to[0,\infty)$ is continuous, strictly increasing function with $K(0)=0$.

We first prove that a minimizer $U$ is piecewise constant in the place $U>g$ or $U<g$.