The theory of [Sobolev spaces](/page/Sobolev%20Space) $W^{1,p}(U)$ provides a powerful framework for studying partial differential equations when $1 < p < \infty$. These spaces are reflexive [Banach spaces](/page/Banach%20Space), and their reflexivity is the engine behind the existence theory: bounded [sequences](/page/Sequence) admit [weakly convergent](/page/Weak%20Convergence) subsequences, which are used to extract solutions via the direct method of the [calculus of variations](/page/Calculus%20of%20Variations). At the endpoint $p = 1$, this machinery breaks down. The space $W^{1,1}(U)$ is not reflexive, and bounded sequences need not have weakly convergent subsequences. Worse, many natural operations — such as taking [limits](/page/Limit) of characteristic functions of smooth domains — produce [functions](/page/Function) whose [distributional](/page/Distribution) [derivatives](/page/Derivative) are not $L^1$ functions at all, but rather *measures*. The space of functions of bounded variation, $BV(U)$, is the correct enlargement of $W^{1,1}(U)$ that accommodates these limits. ## Motivation [motivation] ### Why $W^{1,1}$ Is Too Small Consider a sequence of smooth approximations to the Heaviside step function on $U = (-1, 1)$. Define: \begin{align*} u_\epsilon(x) := \begin{cases} 0 & \text{if } x < -\epsilon, \\ \frac{1}{2} + \frac{x}{2\epsilon} & \text{if } -\epsilon \le x \le \epsilon, \\ 1 & \text{if } x > \epsilon. \end{cases} \end{align*} Each $u_\epsilon$ is Lipschitz (hence in $W^{1,1}(U)$), and its derivative is $u_\epsilon'(x) = \frac{1}{2\epsilon} \mathbb{1}_{(-\epsilon, \epsilon)}(x)$. The $L^1$ norm of the derivative is: \begin{align*} \|u_\epsilon'\|_{L^1(U)} = \int_{-\epsilon}^{\epsilon} \frac{1}{2\epsilon} \, d\mathcal{L}^1(x) = 1 \end{align*} for every $\epsilon > 0$. As $\epsilon \to 0$, the functions $u_\epsilon$ converge in $L^1(U)$ to the Heaviside function $H(x) = \mathbb{1}_{(0,1)}(x)$. The derivatives $u_\epsilon'$ have uniformly bounded $L^1$ norm, yet they converge (in the sense of distributions) not to an $L^1$ function but to the **Dirac measure** $\delta_0$: \begin{align*} \int_{-1}^1 u_\epsilon'(x) \phi(x) \, d\mathcal{L}^1(x) \to \phi(0) = \int_{-1}^1 \phi \, d\delta_0 \qquad \text{for all } \phi \in C^\infty_c(U). \end{align*} Thus $H \notin W^{1,1}(U)$ — its distributional derivative is a measure, not an integrable function — yet $H$ is the $L^1$-limit of $W^{1,1}$ functions with uniformly bounded derivatives. ### The Failure of Weak Compactness at $p = 1$ The root cause is that $L^1(U)$ is not reflexive. The Banach-Alaoglu theorem guarantees that bounded sequences in a reflexive space have weakly convergent subsequences, and this is exactly the mechanism used in the [Difference Quotient Characterisation](/theorems/78) to extract weak derivatives from bounded difference quotients when $p > 1$. At $p = 1$, the bounded sequence $\{u_\epsilon'\}$ does not converge weakly in $L^1$ — it *escapes* to a measure. More precisely, $L^1(U)$ embeds isometrically into the space of finite signed Radon measures $\mathcal{M}(U)$ (which is the dual of $C_0(U)$), and the Banach-Alaoglu theorem applied in $\mathcal{M}(U)$ gives weak-$*$ convergence to a measure. This is the correct topology for extracting limits. ### The Resolution: Allow Measure-Valued Derivatives Instead of insisting that the distributional derivative $Du$ be an $L^1$ function, we enlarge the target space and require only that $Du$ be a **finite Radon measure**. This gives the space $BV(U)$. It is strictly larger than $W^{1,1}(U)$ (since it contains the Heaviside function and characteristic functions of nice [sets](/page/Set)), but it retains enough structure to support compactness theorems that replace the Rellich-Kondrachov theorem at the endpoint $p = 1$. [/motivation] ## Definition [definition: Functions Of Bounded Variation] Let $U \subseteq \mathbb{R}^n$ be an open set. A function $u \in L^1(U)$ is of **bounded variation** if its [distributional derivative](/page/Distributional%20Derivative) is representable by a finite vector-valued Radon measure. That is, $u \in BV(U)$ if there exists a finite $\mathbb{R}^n$-valued Radon measure $Du = (\partial_1 u, \ldots, \partial_n u)$ on $U$ such that for every $i \in \{1, \ldots, n\}$ and every [test function](/page/Test%20Function) $\phi \in C^\infty_c(U)$: \begin{align*} \int_U u \, \partial_i \phi \, d\mathcal{L}^n = -\int_U \phi \, d(\partial_i u). \end{align*} The right-hand side is integration against the signed Radon measure $\partial_i u$, not against an $L^1$ function. [/definition] The key distinction from $W^{1,1}(U)$ is in the nature of $Du$. For Sobolev functions, $Du$ is absolutely [continuous](/page/Continuity) with respect to Lebesgue measure (i.e., $Du = \nabla u \, d\mathcal{L}^n$ for some $\nabla u \in L^1$). For $BV$ functions, $Du$ may have singular components — concentrated on sets of Lebesgue measure zero. [definition: Total Variation Measure] For $u \in BV(U)$, the **total variation measure** $|Du|$ is the non-negative finite Radon measure defined by: \begin{align*} |Du|(A) := \sup \left\{ \sum_{k=1}^\infty |Du(A_k)| : \{A_k\}_{k=1}^\infty \text{ is a Borel partition of } A \right\} \end{align*} for every Borel set $A \subseteq U$. The **total variation** of $u$ on $U$ is the scalar $|Du|(U)$. [/definition] The definition above requires first knowing that $Du$ exists as a measure, and then computing its total variation through Borel partitions. In practice, one wants a criterion that works directly from the function $u$ without presupposing the existence of $Du$. The following result provides exactly this: it characterises $BV$ membership and computes the total variation simultaneously, using only [integration by parts](/theorems/210) against smooth test vector fields. [quotetheorem:591] The power of this characterisation is that the right-hand side involves only $u$ and smooth test fields — no reference to $Du$ is needed. This makes the dual formula the primary tool for verifying $BV$ membership in examples and for proving compactness results, since it provides a uniform bound on the total variation. [definition: BV Norm] The **$BV$ norm** on $BV(U)$ is defined by: \begin{align*} \|u\|_{BV(U)} := \|u\|_{L^1(U)} + |Du|(U). \end{align*} [/definition] A natural question is whether the $BV$ norm makes $BV(U)$ into a complete space. Unlike the Sobolev space $W^{1,1}(U)$, which is a separable Banach space, $BV(U)$ is complete but dramatically larger — so large that it fails to be separable. The completeness is essential for the direct method in the calculus of variations, since one needs the limit of a convergent sequence in $BV$ to remain in $BV$. [quotetheorem:592] The non-[separability](/page/Separable) can be seen from the characteristic functions: for distinct measurable sets $A, B \subset U$ with $\mathcal{L}^n(A \triangle B) > 0$, the distance $\|\mathbb{1}_A - \mathbb{1}_B\|_{BV}$ is bounded below by $\|\mathbb{1}_A - \mathbb{1}_B\|_{L^1} > 0$, and the collection of all such characteristic functions is uncountable with no countable dense subset. ## Examples ### The Heaviside Function [example: Heaviside Function] Let $U = (-1, 1) \subset \mathbb{R}$ and define the Heaviside step function $H(x) := \mathbb{1}_{(0,1)}(x)$. We show that $H \in BV(U)$ with $DH = \delta_0$ and $|DH|(U) = 1$. For any $\phi \in C^\infty_c(U)$, integration by parts gives: \begin{align*} \int_{-1}^1 H(x) \phi'(x) \, d\mathcal{L}^1(x) = \int_0^1 \phi'(x) \, d\mathcal{L}^1(x) = \phi(1) - \phi(0) = -\phi(0), \end{align*} where $\phi(1) = 0$ because $\phi$ has compact support in $(-1,1)$. Thus: \begin{align*} \int_{-1}^1 H \, \phi' \, d\mathcal{L}^1 = -\phi(0) = -\int_{-1}^1 \phi \, d\delta_0. \end{align*} The distributional derivative is $DH = \delta_0$, which is a positive Radon measure with $|DH|(U) = \delta_0(U) = 1$. This function is *not* in $W^{1,1}(U)$: the measure $\delta_0$ is singular with respect to $\mathcal{L}^1$ and cannot be represented as integration against an $L^1$ function. [/example] ### The Cantor Function [example: Cantor Function] The Cantor function $c: [0,1] \to [0,1]$ (also called the devil's staircase) is a continuous, non-decreasing function satisfying $c(0) = 0$, $c(1) = 1$, and $c'(x) = 0$ for $\mathcal{L}^1$-a.e. $x \in [0,1]$ (since $c$ is constant on each of the open intervals that make up the complement of the Cantor set $\mathcal{C}$, which has full measure $\mathcal{L}^1([0,1] \setminus \mathcal{C}) = 1$). The Cantor function belongs to $BV(0,1)$ with $|Dc|(0,1) = 1$: the total variation equals the total increase of $c$, which is $c(1) - c(0) = 1$. However, the distributional derivative $Dc$ is the **Cantor measure** — a probability measure supported on the Cantor set $\mathcal{C}$ (which has $\mathcal{L}^1(\mathcal{C}) = 0$). The Cantor measure is: - singular with respect to $\mathcal{L}^1$ (concentrated on a null set), so $c \notin W^{1,1}(0,1)$, - continuous (it assigns zero mass to every singleton $\{x\}$), so it has no jump part. This demonstrates that $BV$ functions can have derivatives that are singular continuous — neither absolutely continuous nor concentrated on countably many points. [/example] ### Characteristic Functions and Perimeter [example: Characteristic Function Of A Smooth Set] Let $E \subset \mathbb{R}^n$ be a bounded [open set](/page/Open%20Set) with $C^1$ [boundary](/page/Boundary) $\partial E$, and let $U \supseteq \bar{E}$ be an open set. The characteristic function $\mathbb{1}_E$ is in $BV(U)$, and its distributional derivative is the vector-valued surface measure: \begin{align*} D\mathbb{1}_E = -\nu_E \, \mathcal{H}^{n-1} \lfloor \partial E, \end{align*} where $\nu_E$ is the outward unit normal to $\partial E$ and $\mathcal{H}^{n-1} \lfloor \partial E$ denotes $(n-1)$-dimensional [Hausdorff measure](/page/Hausdorff%20Measure) restricted to $\partial E$. To verify this, let $\Phi \in C^\infty_c(U; \mathbb{R}^n)$. By the divergence theorem: \begin{align*} \int_U \mathbb{1}_E \operatorname{div} \Phi \, d\mathcal{L}^n = \int_E \operatorname{div} \Phi \, d\mathcal{L}^n = \int_{\partial E} \Phi \cdot \nu_E \, d\mathcal{H}^{n-1}. \end{align*} The total variation is: \begin{align*} |D\mathbb{1}_E|(U) = \mathcal{H}^{n-1}(\partial E), \end{align*} which is exactly the **perimeter** of $E$. This connection between $BV$ and geometry — the total variation of a characteristic function equals the surface area of the boundary — is the foundation of De Giorgi's theory of sets of finite perimeter. [/example] ## Relationship to Sobolev Spaces ### $W^{1,1}(U) \subset BV(U)$: Strict Inclusion The Sobolev space $W^{1,1}(U)$ sits inside $BV(U)$ as a proper closed subspace. The following result makes the relationship precise and provides a clean criterion for when a $BV$ function is actually Sobolev. [quotetheorem:593] The inclusion is strict: the Heaviside function and the Cantor function are both in $BV$ but not in $W^{1,1}$. These examples are not pathological — they represent the two distinct mechanisms by which $Du$ can fail to be absolutely continuous. The Heaviside function has a jump discontinuity (the singular part of $Du$ is concentrated on a single point), while the Cantor function is continuous but its derivative is spread over a set of Lebesgue measure zero. The structure theorem below makes this decomposition precise. ### Why $BV$ Is the Natural Endpoint The Sobolev embedding $W^{1,p}(U) \hookrightarrow L^{p^*}(U)$ for $p < n$ degenerates as $p \to 1$: the Sobolev conjugate $p^* = np/(n-p) \to n/(n-1) =: 1^*$. The question is whether $BV$ inherits the same embedding at $p = 1$. This is not automatic, since $BV$ functions are more singular than $W^{1,1}$ functions, but the embedding survives because only the total mass of $Du$ — not its regularity — controls the $L^{1^*}$ norm. [quotetheorem:594] This is the exact $p = 1$ analogue of the Gagliardo-Nirenberg-Sobolev inequality. The proof for smooth functions is identical (using the same representation formula and Luzin's inequality), and extends to $BV$ by approximation. The exponent $n/(n-1)$ is sharp. ## Structure of the Derivative The distributional derivative $Du$ of a $BV$ function can be decomposed into parts with distinct geometric characters. This decomposition reveals the fine structure of $BV$ functions and is fundamental to the regularity theory for variational problems. ### Radon-Nikodym Decomposition Since $Du$ is a finite vector-valued Radon measure and $\mathcal{L}^n$ is a $\sigma$-finite positive measure, the Radon-Nikodym theorem decomposes $Du$ into absolutely continuous and singular parts: \begin{align*} Du = D^a u + D^s u, \end{align*} where $D^a u \ll \mathcal{L}^n$ (absolutely continuous) and $D^s u \perp \mathcal{L}^n$ (singular). The absolutely continuous part has a density: \begin{align*} D^a u = \nabla u \, d\mathcal{L}^n, \end{align*} where $\nabla u \in L^1(U; \mathbb{R}^n)$ is the **approximate gradient** of $u$. This is the "Sobolev part" of the derivative — the part that resembles a classical gradient. ### The Jump Set and Cantor Part The singular part $D^s u$ can be further decomposed by examining the geometry of the set where $u$ is discontinuous. The following definition and subsequent structure theorem make this precise. [definition: Jump Set] Let $u \in BV(U)$. The **jump set** $J_u$ is the set of points $x \in U$ where $u$ has an approximate jump discontinuity: there exist values $u^+(x) \neq u^-(x)$ and a unit normal $\nu_u(x) \in \mathbb{S}^{n-1}$ such that \begin{align*} \lim_{r \to 0} \frac{1}{\mathcal{L}^n(B_r(x))} \int_{B_r(x) \cap \{y : (y-x) \cdot \nu_u > 0\}} |u(y) - u^+(x)| \, d\mathcal{L}^n(y) = 0, \end{align*} and similarly for $u^-$ on the opposite half-ball $\{y : (y - x) \cdot \nu_u < 0\}$. [/definition] The jump set captures the "hypersurface-like" discontinuities of $u$ — the places where $u$ has two distinct approximate limits from either side of a codimension-one surface. A key fact, which is part of the structure theorem below, is that $J_u$ has a very controlled geometry: it is not an arbitrary subset of $U$ but is concentrated on countably many $C^1$ hypersurfaces, up to a negligible set. This regularity of the discontinuity set is one of the deepest aspects of $BV$ theory. The full decomposition of $Du$ separates three mutually singular components, each with a distinct geometric character: the absolutely continuous part (Sobolev-like, diffuse), the jump part (concentrated on codimension-one surfaces), and the Cantor part (singular with respect to Lebesgue measure but not concentrated on any rectifiable set). [quotetheorem:595] The structure theorem is one of the crown jewels of geometric measure theory. Part (i) shows that the discontinuity set of a $BV$ function, while potentially complicated, is always "essentially $(n-1)$-dimensional" — it lives on countably many smooth hypersurfaces. The Cantor part $D^c u$ in (ii) is the most mysterious component: it is singular with respect to both Lebesgue measure and the $(n-1)$-dimensional Hausdorff measure on $J_u$. The Cantor function on $[0,1]$ is the canonical example: its entire distributional derivative is pure Cantor part, supported on the Cantor set (which has Hausdorff dimension $\log 2 / \log 3$, strictly between $0$ and $1$). Each of our earlier examples isolates one component: the Heaviside function has $Du = D^j u = \delta_0$ (pure jump), the Cantor function has $Du = D^c u$ (pure Cantor part), and any $W^{1,1}$ function has $Du = D^a u = \nabla u \, d\mathcal{L}^n$ (purely absolutely continuous). A general $BV$ function is a superposition of all three behaviours. ## Compactness The most important property of $BV(U)$ for applications to the calculus of variations is its compactness theorem, which replaces the Rellich-Kondrachov theorem at the endpoint $p = 1$. ### Why Rellich-Kondrachov Fails at $p = 1$ The Rellich-Kondrachov theorem states that $W^{1,p}(U) \hookrightarrow \hookrightarrow L^q(U)$ for $q < p^*$ when $1 < p < n$. At $p = 1$, the space $W^{1,1}(U)$ still embeds compactly into $L^1(U)$, but the problem is different: many natural minimisation sequences leave $W^{1,1}$ entirely. A sequence of smooth functions with uniformly bounded $W^{1,1}$ norms may converge in $L^1$ to a function that is only in $BV$. To capture these limits, we need compactness in $BV$ itself. [quotetheorem:596] The convergence is **strong in $L^1$**, not merely weak. This is stronger than what one might expect — it comes from the fact that $BV$ functions satisfy a Poincaré-type inequality that prevents oscillations from developing without increasing the total variation. An important companion to the compactness theorem is the fact that the total variation functional is lower semicontinuous — it cannot increase in the limit, but it can decrease. This semicontinuity is what allows the direct method to work: one extracts a convergent subsequence and then verifies that the limit has controlled total variation. [quotetheorem:597] The inequality can be strict: the limit function $u$ may have strictly smaller total variation than the sequence. Mass in the derivatives can "escape" by cancellation or by concentrating on sets of measure zero that contribute differently in the limit. For instance, consider oscillatory approximations to a constant function — the derivatives may have large total variation, but they cancel in the limit. ### The Direct Method in $BV$ The compactness and lower semicontinuity theorems together form the foundation of the direct method for variational problems in $BV$. To minimise a functional of the form: \begin{align*} \mathcal{F}(u) = \int_U f(\nabla u) \, d\mathcal{L}^n + \int_U g(u^+ - u^-) \, d\mathcal{H}^{n-1} \lfloor J_u + |D^c u|(U), \end{align*} one takes a minimising sequence, extracts a convergent subsequence in $L^1$ by the compactness theorem, and then uses lower semicontinuity to conclude that the limit is a minimiser. Functionals of this form arise naturally in image processing (the Mumford-Shah functional), fracture mechanics (Griffith's model), and phase transitions (the Modica-Mortola functional). ## The Coarea Formula The coarea formula for $BV$ functions generalises the [Coarea Formula (Classical)](/theorems/24) from Lipschitz functions to $BV$, and provides a powerful link between the total variation of a function and the perimeters of its level sets. [quotetheorem:598] This identity says that the total variation of $u$ equals the integral of the perimeters of all its level sets. It is remarkable because it reduces a statement about the derivative of a function to a purely geometric statement about the boundaries of its superlevel sets. [example: Coarea Formula For The Heaviside Function] For the Heaviside function $H = \mathbb{1}_{(0,1)}$ on $U = (-1,1)$, the superlevel sets are: \begin{align*} \{H > t\} = \begin{cases} (-1, 1) & \text{if } t < 0, \\ (0, 1) & \text{if } 0 \le t < 1, \\ \emptyset & \text{if } t \ge 1. \end{cases} \end{align*} The perimeters are: $\operatorname{Per}(\{H > t\}; U) = 0$ for $t < 0$ (no boundary inside $U$) and $t \ge 1$ (empty set), and $\operatorname{Per}(\{H > t\}; U) = 1$ for $0 \le t < 1$ (the single point $\{0\}$ contributes $\mathcal{H}^0(\{0\}) = 1$). Thus: \begin{align*} \int_{-\infty}^{\infty} \operatorname{Per}(\{H > t\}; U) \, d\mathcal{L}^1(t) = \int_0^1 1 \, d\mathcal{L}^1(t) = 1 = |DH|(U). \end{align*} [/example] ## Sets of Finite Perimeter The connection between $BV$ functions and geometry is deepest for characteristic functions. [definition: Set Of Finite Perimeter] A Lebesgue measurable set $E \subseteq \mathbb{R}^n$ has **finite perimeter** in $U$ if $\mathbb{1}_E \in BV(U)$. The **perimeter** of $E$ in $U$ is: \begin{align*} \operatorname{Per}(E; U) := |D\mathbb{1}_E|(U) = \sup \left\{ \int_E \operatorname{div} \Phi \, d\mathcal{L}^n : \Phi \in C^\infty_c(U; \mathbb{R}^n), \; |\Phi| \le 1 \right\}. \end{align*} [/definition] For sets with smooth boundary, $\operatorname{Per}(E; U) = \mathcal{H}^{n-1}(\partial E \cap U)$. But the definition extends far beyond smooth sets: any measurable set whose boundary is not too wild (in the sense that the divergence theorem still holds in a generalised form) has finite perimeter. The following theorem, due to De Giorgi, reveals the geometric structure of sets of finite perimeter and shows that a generalised divergence theorem holds for them. [quotetheorem:599] This is a far-reaching generalisation of the classical divergence theorem. It asserts that even for sets with highly irregular [topological](/page/Topology) boundaries, the "measure-theoretic boundary" $\partial^* E$ is a rectifiable set — it can be covered, up to an $\mathcal{H}^{n-1}$-null set, by countably many $C^1$ hypersurfaces — and the distributional derivative of $\mathbb{1}_E$ is precisely the surface measure weighted by the outward normal. The essential boundary $\partial^* E$ may be strictly smaller than the topological boundary $\partial E$: sets can have topological boundary of positive Lebesgue measure (like the boundary of a fat Cantor set) while the essential boundary remains $(n-1)$-dimensional. ### The Isoperimetric Inequality in $BV$ The classical isoperimetric inequality — among all sets of fixed volume, the ball has the smallest perimeter — admits a clean formulation in the $BV$ framework. The inequality is remarkable because it makes no regularity assumptions on $E$: even sets with fractal boundaries satisfy it, provided the perimeter is finite. [quotetheorem:600] This inequality is equivalent to the Sobolev inequality for $W^{1,1}$ (take $u = \mathbb{1}_E$ and use the coarea formula). The $BV$ formulation makes the geometric content transparent: controlling the boundary (perimeter) controls the bulk (volume). ## [Fundamental Theorem of Calculus](/theorems/632) for $BV$ Functions In classical analysis, the fundamental theorem of calculus states that a function with an [integrable](/page/Integral) derivative can be recovered (up to a constant) by integrating its derivative. A natural question is whether an analogous result holds for $BV$ functions: given the distributional derivative $Du$ — which is now a measure rather than an integrable function — can one recover $u$ from $Du$? The answer depends dramatically on the dimension. ### The One-Dimensional Case In one dimension, the situation is completely clean. Every finite signed Radon measure on an interval is the distributional derivative of a $BV$ function, and the recovery formula is the obvious generalisation of the classical fundamental theorem — one simply replaces the [Lebesgue integral](/page/Lebesgue%20Integral) with integration against the measure $Du$. [quotetheorem:601] The first part says that every one-dimensional $BV$ function can be recovered from its derivative measure, just as a classical $C^1$ function is recovered from its derivative by integration. The second part is the converse: every finite measure on an interval *is* the derivative of some $BV$ function. Together, they establish a bijective correspondence between $BV(I)$ (modulo constants) and the space of finite signed Radon measures $\mathcal{M}(I)$. This is a powerful result because it means that in one dimension, there is no obstruction to being a $BV$ derivative: any finite measure will do. The three components of the Structure Theorem correspond to familiar objects in one dimension. The absolutely continuous part $D^a u = u' \, d\mathcal{L}^1$ is the classical derivative (existing a.e.). The jump part $D^j u = \sum_{k} (u(x_k^+) - u(x_k^-)) \delta_{x_k}$ records the jumps of $u$ at its countably many discontinuities. The Cantor part $D^c u$ captures the remaining singular continuous behaviour, as exhibited by the Cantor function. ### The Higher-Dimensional Case: The Curl Condition In dimensions $n \ge 2$, the situation is fundamentally different. Not every $\mathbb{R}^n$-valued finite Radon measure on $U$ arises as the distributional derivative of a $BV$ function. The obstruction is a compatibility condition on the components of the measure — a generalisation of the classical fact that a vector field $F = (F_1, \ldots, F_n)$ is a gradient if and only if it is curl-free. To see why an obstruction must arise, recall that if $u \in BV(U)$, then $Du = (\partial_1 u, \ldots, \partial_n u)$ where each $\partial_i u$ is a scalar-valued measure. Since these measures all come from a single scalar function $u$, they satisfy the distributional symmetry condition $\partial_i(\partial_j u) = \partial_j(\partial_i u)$ — the mixed partials commute. For measures, this translates into a constraint: the vector-valued measure $\mu = (\mu_1, \ldots, \mu_n)$ must be **curl-free** in the distributional sense. [quotetheorem:602] The condition $\partial_i \mu_j = \partial_j \mu_i$ is the exact measure-theoretic analogue of the condition $\operatorname{curl} F = 0$ for smooth vector fields. When $n = 2$, there is a single condition: $\partial_1 \mu_2 = \partial_2 \mu_1$. When $n = 3$, there are three conditions, corresponding to the vanishing of each component of $\operatorname{curl} \mu$. The contrast with the one-dimensional case is instructive. In one dimension, the "curl" condition is vacuous — there is only one component, so there is no compatibility to check. This is why every finite measure on an interval is a $BV$ derivative. In two or more dimensions, the condition is a genuine constraint: one cannot freely prescribe the $n$ component measures $\mu_1, \ldots, \mu_n$ independently. They must be related by the commutativity of mixed partial derivatives. [example: Failure Of The Curl Condition] Let $U = B(0,1) \subset \mathbb{R}^2$ and define the $\mathbb{R}^2$-valued measure $\mu = (\mu_1, \mu_2)$ by $\mu_1 = \delta_0$ and $\mu_2 = 0$. To check the curl condition, take any $\phi \in C^\infty_c(U)$: \begin{align*} \int_U \partial_1 \phi \, d\mu_2 &= 0, \\ \int_U \partial_2 \phi \, d\mu_1 &= \partial_2 \phi(0). \end{align*} These are not equal in general (choose $\phi$ with $\partial_2 \phi(0) \neq 0$), so $\mu$ is not curl-free. Therefore, there is no $u \in BV(U)$ with $Du = (\delta_0, 0)$. Geometrically, this makes sense: the measure $(\delta_0, 0)$ represents a "derivative" that points purely in the $x_1$-direction and is concentrated at the origin. If this were the gradient of some $u$, then $u$ would have a jump discontinuity at the origin in the $x_1$-direction but no variation in the $x_2$-direction. However, a scalar function cannot have a codimension-two singularity of this type — jumps in $BV$ functions occur along codimension-one surfaces (as guaranteed by the Structure Theorem), not at isolated points. [/example] The simple connectivity hypothesis in the theorem is essential. On a multiply connected domain, additional topological obstructions arise: a curl-free measure may fail to be a gradient because it has nontrivial "periods" around holes in the domain, just as in the smooth case where a curl-free vector field on an annulus need not be a gradient (the classical example being $(-y/(x^2+y^2), x/(x^2+y^2))$ on $\mathbb{R}^2 \setminus \{0\}$). ## Problems [problem] Let $U = (0,1) \subset \mathbb{R}$ and define the function $u: U \to \mathbb{R}$ by: \begin{align*} u(x) = \sin\left(\frac{1}{x}\right). \end{align*} Determine whether $u \in BV(0,1)$. [/problem] [solution] **Step 1: Classical derivative away from the singularity.** For $x \in (0,1)$, the function $u$ is smooth with classical derivative: \begin{align*} u'(x) = -\frac{1}{x^2} \cos\left(\frac{1}{x}\right). \end{align*} **Step 2: Compute the total variation via the $L^1$ norm of the derivative.** Since $u$ is $C^1$ on $(0,1)$ and its classical derivative is locally integrable, $u \in BV(0,1)$ if and only if $u' \in L^1(0,1)$ (for smooth functions on an interval, the distributional derivative coincides with the classical one, and the total variation equals $\|u'\|_{L^1}$). We compute: \begin{align*} \int_0^1 |u'(x)| \, d\mathcal{L}^1(x) = \int_0^1 \frac{1}{x^2} \left|\cos\left(\frac{1}{x}\right)\right| \, d\mathcal{L}^1(x). \end{align*} Substitute $t = 1/x$, so $dt = -1/x^2 \, dx = -t^2 \, dx$, giving $dx = dt/t^2$. The limits change: $x = 1 \mapsto t = 1$ and $x \to 0^+ \mapsto t \to \infty$. Thus: \begin{align*} \int_0^1 \frac{1}{x^2} \left|\cos\left(\frac{1}{x}\right)\right| \, d\mathcal{L}^1(x) = \int_1^{\infty} t^2 |\cos(t)| \cdot \frac{1}{t^2} \, d\mathcal{L}^1(t) = \int_1^{\infty} |\cos(t)| \, d\mathcal{L}^1(t). \end{align*} **Step 3: Divergence of the integral.** The function $|\cos(t)|$ is periodic with period $\pi$ and has mean value: \begin{align*} \frac{1}{\pi} \int_0^{\pi} |\cos(t)| \, d\mathcal{L}^1(t) = \frac{2}{\pi} > 0. \end{align*} Therefore: \begin{align*} \int_1^{N\pi} |\cos(t)| \, d\mathcal{L}^1(t) \ge (N-1) \cdot 2 - C \to \infty \quad \text{as } N \to \infty. \end{align*} **Conclusion.** The total variation $|Du|(0,1) = \int_0^1 |u'| \, d\mathcal{L}^1 = \infty$, so $u = \sin(1/x) \notin BV(0,1)$. [/solution] [problem] Let $U = B(0,1) \subset \mathbb{R}^n$ (the open unit ball) and let $E = \{x \in U : x_1 > 0\}$ be the right half-ball. Compute $\operatorname{Per}(E; U)$. [/problem] [solution] **Step 1: Identify the essential boundary.** The set $E$ has piecewise smooth boundary. Within $U$, the boundary $\partial E \cap U$ is the flat disk: \begin{align*} \partial E \cap U = \{x \in U : x_1 = 0\} = B'(0,1) \times \{0\}, \end{align*} where $B'(0,1) \subset \mathbb{R}^{n-1}$ is the unit ball in the remaining coordinates. The outward unit normal (pointing out of $E$, i.e., in the $-e_1$ direction) is $\nu_E = -e_1$. **Step 2: Apply the divergence theorem formula.** Since $E$ has Lipschitz boundary relative to $U$, the perimeter equals: \begin{align*} \operatorname{Per}(E; U) = \mathcal{H}^{n-1}(\partial E \cap U) = \mathcal{H}^{n-1}(B'(0,1)). \end{align*} The $(n-1)$-dimensional Hausdorff measure of the unit ball in $\mathbb{R}^{n-1}$ is its Lebesgue measure: \begin{align*} \mathcal{H}^{n-1}(B'(0,1)) = \mathcal{L}^{n-1}(B(0,1) \subset \mathbb{R}^{n-1}) = \omega_{n-1}, \end{align*} where $\omega_{n-1} = \frac{\pi^{(n-1)/2}}{\Gamma((n-1)/2 + 1)}$ is the volume of the unit ball in $\mathbb{R}^{n-1}$. **Step 3: Verification via the dual characterisation.** We verify using the sup-formula. For any $\Phi \in C^\infty_c(U; \mathbb{R}^n)$ with $|\Phi| \le 1$: \begin{align*} \int_E \operatorname{div} \Phi \, d\mathcal{L}^n = \int_{\partial E \cap U} \Phi \cdot (-e_1) \, d\mathcal{H}^{n-1} = -\int_{B'(0,1)} \Phi_1(0, x') \, d\mathcal{L}^{n-1}(x'). \end{align*} The supremum over $|\Phi| \le 1$ is achieved by taking $\Phi = -e_1$ near $\{x_1 = 0\}$, giving: \begin{align*} \operatorname{Per}(E; U) = \int_{B'(0,1)} 1 \, d\mathcal{L}^{n-1}(x') = \omega_{n-1}. \end{align*} **Conclusion.** $\operatorname{Per}(E; U) = \omega_{n-1}$. [/solution] ## References - Ambrosio, L., Fusco, N., and Pallara, D., *Functions of Bounded Variation and Free Discontinuity Problems* (2000). - Evans, L. C. and Gariepy, R. F., *Measure Theory and Fine Properties of Functions* (2015). - Giusti, E., *Minimal Surfaces and Functions of Bounded Variation* (1984).