BV function

Functions of bounded variation, $BV$ functions, are functions whose distributional derivative is a finite Radon measure. More precisely:

Definition 1 (functions of bounded variation).

Let $\mathrm{\Omega }\mathrm{\subset }{\mathrm{R}}^n$ be an open set. We say that a function $u\mathrm{\in }{L}^{\mathrm{1}}\mathit{}\mathrm{(}\mathrm{\Omega }\mathrm{)}$has bounded variation, and write $u\mathrm{\in }B\mathit{}V\mathit{}\mathrm{(}\mathrm{\Omega }\mathrm{)}$, if there exists a finite Radon vector measure $D\mathit{}u\mathrm{\in }\mathrm{M}\mathit{}\mathrm{(}\mathrm{\Omega }\mathrm{,}{\mathrm{R}}^n\mathrm{)}$ such that

$${\int }_{\mathrm{\Omega }}u(x)\mathrm{div}\varphi (x)𝑑x=-{\int }_{\mathrm{\Omega }}⟨\varphi (x),Du(x)⟩$$

for every function $\varphi \mathrm{\in }{C}_c^{\mathrm{1}}\mathit{}\mathrm{(}\mathrm{\Omega }\mathrm{,}{\mathrm{R}}^n\mathrm{)}$. The measure $D\mathit{}u$,represents the distributional derivative of $u$ since the above equality holds true for every $\varphi \mathrm{\in }{C}_c^{\mathrm{\infty }}\mathit{}\mathrm{(}\mathrm{\Omega }\mathrm{,}{\mathrm{R}}^n\mathrm{)}$.

Notice that $W^{1,1}(\mathrm{\Omega })\subset BV(\mathrm{\Omega })$. In fact if $u\in W^{1,1}(\mathrm{\Omega })$ one can choose $\mu :=\nabla uℒ$ (where $ℒ$ is the Lebesgue measure on $\mathrm{\Omega }$).The equality $\int u\mathrm{div}\varphi =-\int \varphi 𝑑\mu =-\int \varphi \nabla u$is nothing else than the definition of weak derivative, and hence holds true.One can easily find an example of a $BV$ functions which is not $W^{1,1}$.

An equivalent definition can be given as follows.

Definition 2 (variation).

Given $u\mathrm{\in }{L}^{\mathrm{1}}\mathit{}\mathrm{(}\mathrm{\Omega }\mathrm{)}$ we define the variation of $u$ in $\mathrm{\Omega }$ as

$$V(u,\mathrm{\Omega }):=sup\{{\int }_{\mathrm{\Omega }}u\mathrm{div}\varphi :\varphi \in {𝒞}_c^1(\mathrm{\Omega },{ℝ}^n),{\parallel \varphi \parallel }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le 1\}.$$

We define .

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