This course extends the measure-theoretic foundations of Geometric Measure Theory I and II to study functions and sets with minimal regularity assumptions. Building on the Hausdorff measures and coarea formulas developed in earlier volumes, we investigate Sobolev spaces and BV (bounded variation) functions—classes where classical derivatives need not exist, yet a meaningful notion of weak differentiability and integration by parts persists. The central achievement is De Giorgi's structure theorem for sets of finite perimeter, which reveals that the perimeter measure, though defined variationally through competitors of minimal area, coincides with the classical surface area on the reduced boundary. This remarkable confluence of analytic and geometric perspectives culminates in the Gauss-Green theorem for non-smooth domains, extending the fundamental theorem of calculus to domains whose boundaries are merely Hausdorff-measurable.

The course is organized in three natural phases. Chapters 1–7 develop Sobolev space theory, introducing weak derivatives, approximation by smooth functions, trace operators, extension theorems, and Sobolev inequalities that govern the interplay between integrability and regularity. This framework provides the analytic machinery needed for the study of lower-regularity objects. Chapters 8–11 parallel this development for BV functions: we define BV as the space of $L^1$ functions whose distributional derivatives are measures, establish the structure theorem showing any BV function is a sum of a jump discontinuity and a Cantor part, prove approximation and compactness results, and derive the coarea formula in the BV setting. Chapters 12–14 form the geometric core, where De Giorgi's theorem appears: sets of finite perimeter are characterized by having a *reduced boundary* of finite Hausdorff measure, and the perimeter equals the $(n-1)$-dimensional measure of this reduced boundary almost everywhere. The Gauss-Green theorem follows, allowing divergence identities to hold for such rough domains. Finally, Chapter 15 grounds this abstract theory with concrete examples—explicit computations for fractals, corners, and cusps—and worked problems that reinforce the interplay between analytic regularity and geometric structure.

Throughout, a unifying theme emerges: regularity and measure are not fixed properties but *learnable* from the behavior of the function or set itself. The machinery of weak derivatives and variational characterizations permits rigorous analysis of objects far rougher than classical smooth manifolds, yet rich enough to support integration by parts and related geometric theorems. This volume solidifies the transition from smooth differential geometry to the rectifiable geometry of rough sets, equipping students with tools to work effectively in geometric analysis, partial differential equations, and the calculus of variations.

# 1. Sobolev Spaces: Definitions and Basic Properties

*This chapter lays the functional-analytic foundation for the entire course. Classical derivatives are too rigid for variational problems and PDEs: solutions need not be smooth, yet we require some notion of differentiation to make sense of equations and energy functionals. The weak derivative — defined by integration by parts against test functions — extends classical differentiation to $L^p$ functions, and the Sobolev spaces $W^{1,p}(\Omega)$ collect those $L^p$ functions whose weak gradient is again $L^p$. These spaces are the natural setting for existence theory via the direct method, and their analysis here will be used in every subsequent chapter, culminating in the BV spaces of Chapter 8.*

## Weak Derivatives and the Need for Generalised Differentiation

Classical analysis demands pointwise derivatives, but variational problems naturally produce functions that fail to be classically differentiable at many points. A striking example is the absolute value function $u(x) = |x|$ on $\mathbb{R}$: it is Lipschitz continuous but not differentiable at the origin. Despite this, it has a perfectly well-defined derivative almost everywhere — the sign function — and this derivative captures all the information needed for integration and variational analysis. The theory of weak derivatives formalises this idea by insisting only that integration by parts holds, rather than demanding a pointwise limit.

[definition: Weak Derivative] Let $\Omega \subset \mathbb{R}^n$ be open, let $u \in L^1_{\mathrm{loc}}(\Omega)$, and let $\alpha = (\alpha_1, \dots, \alpha_n)$ be a multi-index with $|\alpha| = \alpha_1 + \cdots + \alpha_n$. A function $v \in L^1_{\mathrm{loc}}(\Omega)$ is the **$\alpha$-th weak derivative** of $u$, written $v = D^\alpha u$, if \begin{align*} \int_\Omega u \, D^\alpha \phi \, d\mathcal{L}^n = (-1)^{|\alpha|} \int_\Omega v \, \phi \, d\mathcal{L}^n \end{align*} for every test function $\phi \in C_c^\infty(\Omega)$. [/definition]

The formula is simply integration by parts, with the boundary terms absent because $\phi$ has compact support. This observation immediately shows that if $u$ is classically $C^{|\alpha|}$, then its classical $\alpha$-th derivative is also its weak derivative — the two notions are consistent.

The weak derivative, when it exists, is unique up to sets of $\mathcal{L}^n$-measure zero. To see why, suppose both $v$ and $w$ satisfy the integration-by-parts identity. Then $\int_\Omega (v - w) \phi \, d\mathcal{L}^n = 0$ for all $\phi \in C_c^\infty(\Omega)$, which forces $v = w$ $\mathcal{L}^n$-a.e. by the fundamental lemma of the calculus of variations.

<!-- illustration-needed: 1D plot of |x| on [-1,1] showing the kink at the origin, alongside its weak derivative (sign function) — emphasises the contrast between classical and weak differentiability -->

[example: Weak Derivative of $|x|$] Consider $u(x) = |x|$ on $\Omega = (-1, 1) \subset \mathbb{R}$. Define $v(x) = \operatorname{sgn}(x)$, i.e., $v(x) = 1$ for $x > 0$ and $v(x) = -1$ for $x < 0$. For any $\phi \in C_c^\infty(-1, 1)$, \begin{align*} \int_{-1}^1 |x| \, \phi'(x) \, d\mathcal{L}^1 &= \int_{-1}^0 (-x) \phi'(x) \, d\mathcal{L}^1 + \int_0^1 x \, \phi'(x) \, d\mathcal{L}^1. \end{align*} Integrating each piece by parts: \begin{align*} \int_{-1}^0 (-x) \phi'(x) \, d\mathcal{L}^1 &= \left[(-x)\phi(x)\right]_{-1}^0 + \int_{-1}^0 \phi(x) \, d\mathcal{L}^1 = 0 + \phi(-1)\cdot(-1)\cdot(-1) + \int_{-1}^0 \phi(x) \, d\mathcal{L}^1, \\ \int_0^1 x \, \phi'(x) \, d\mathcal{L}^1 &= \left[x\phi(x)\right]_0^1 - \int_0^1 \phi(x) \, d\mathcal{L}^1 = -\int_0^1 \phi(x) \, d\mathcal{L}^1. \end{align*} The boundary terms at $x = \pm 1$ vanish because $\phi$ has compact support in $(-1,1)$, so $\phi(\pm 1) = 0$. The boundary terms at $x = 0$ from the two integrations cancel: the left piece contributes $-\phi(0) \cdot (-1) = \phi(0)$ and the right piece contributes $-\phi(0) \cdot (+1) = -\phi(0)$; these are equal and opposite, so they sum to zero. Adding the remaining integrals: $\int_{-1}^1 |x| \phi'(x) \, d\mathcal{L}^1 = \int_{-1}^0 \phi \, d\mathcal{L}^1 - \int_0^1 \phi \, d\mathcal{L}^1 = -\int_{-1}^1 \operatorname{sgn}(x)\, \phi(x) \, d\mathcal{L}^1$. Thus $D^1 u = \operatorname{sgn}(x)$, and this weak derivative exists despite $u$ not being classically differentiable at $0$. [/example]

The example above deals only with first-order differentiation, but the definition extends naturally to all multi-indices. Before moving on, it is worth recording what higher-order weak derivatives look like and how they relate to membership in Sobolev spaces.

[remark: Higher-Order Weak Derivatives] The definition extends naturally to all orders. For $u \in L^1_{\mathrm{loc}}(\Omega)$, the weak derivative $D^\alpha u$ is the weak partial derivative of order $|\alpha|$. When $|\alpha| = 1$ and we consider all first-order derivatives, we write the weak gradient as $\nabla u = (D_{e_1} u, \dots, D_{e_n} u)$, where $D_{e_i} u$ is the weak partial derivative in the $i$-th coordinate direction. The weak gradient, when it exists as an $L^p$ function, is what defines membership in the Sobolev space $W^{1,p}$. [/remark]

Not every function has a weak derivative. The key obstruction is that discontinuities concentrated on sets of positive $(n-1)$-dimensional measure prevent integration by parts from holding globally. For example, the [Heaviside function](/page/Heaviside%20Function) $u = \mathbb{1}_{(0,1)}$ on $(-1,1)$ does not have a weak derivative in $L^1_{\mathrm{loc}}$: if $v$ were such a derivative, then integrating by parts would force $\int \mathbb{1}_{(0,1)} \phi' \, d\mathcal{L}^1 = -\int v\phi \, d\mathcal{L}^1$ for all $\phi \in C_c^\infty$, but the left side computes to $\phi(0) - \phi(1) = \phi(0)$ (since $\phi(1)=0$), which is the Dirac delta evaluated at $0$ — not representable by any $L^1_{\mathrm{loc}}$ function.

## The Sobolev Space $W^{1,p}(\Omega)$

To see why $L^p$ alone is insufficient, consider minimising an energy like $\int_\Omega |\nabla u|^p \, d\mathcal{L}^n$ over $L^p(\Omega)$. A minimising sequence might converge in $L^p$ to some limit $u$, but we have no guarantee that the limit has a gradient at all — the gradients of the approximating sequence might diverge, or the limit might be a function whose gradient exists only in a distributional sense outside $L^p$. What is needed is a space that keeps $u$ and its gradient $\nabla u$ together and treats both as $L^p$ objects simultaneously. The Sobolev space $W^{1,p}(\Omega)$ does exactly this.

[definition: Sobolev Space $W^{1,p}$] Let $\Omega \subset \mathbb{R}^n$ be open and let $1 \le p \le \infty$. The **Sobolev space** $W^{1,p}(\Omega)$ is \begin{align*} W^{1,p}(\Omega) := \{ u \in L^p(\Omega) : D_{e_i} u \in L^p(\Omega) \text{ for } i = 1, \dots, n \}, \end{align*} equipped with the norm \begin{align*} \|u\|_{W^{1,p}(\Omega)} := \|u\|_{L^p(\Omega)} + \|\nabla u\|_{L^p(\Omega)}, \end{align*} where $\|\nabla u\|_{L^p(\Omega)}^p := \sum_{i=1}^n \|D_{e_i} u\|_{L^p(\Omega)}^p$ for $p < \infty$, and $\|\nabla u\|_{L^\infty(\Omega)} := \max_i \|D_{e_i} u\|_{L^\infty(\Omega)}$. [/definition]

The norm bundles the $L^p$ size of $u$ and its weak gradient into a single quantity; convergence in $W^{1,p}$ means convergence of both the function and its gradient simultaneously. **How to verify membership in practice:** to show $u \in W^{1,p}(\Omega)$, one checks (i) $u \in L^p(\Omega)$, (ii) computes the candidate weak gradient $v$ from classical formulas (wherever $u$ is smooth), then (iii) verifies $\int_\Omega u \, \partial_i \phi \, d\mathcal{L}^n = -\int_\Omega v_i \phi \, d\mathcal{L}^n$ for all $\phi \in C_c^\infty(\Omega)$. A common pitfall: the classical derivative computed pointwise a.e. is not always the weak derivative — the Cantor staircase (discussed below) has classical derivative zero a.e. yet is not in $W^{1,1}$, because the integration-by-parts identity fails.

[remark: Equivalent Norms] The norm $\|u\|_{W^{1,p}} = \|u\|_{L^p} + \|\nabla u\|_{L^p}$ is equivalent to $(\|u\|_{L^p}^p + \|\nabla u\|_{L^p}^p)^{1/p}$ for $1 \le p < \infty$, and both are standard in the literature. We use the additive form throughout, following Evans–Gariepy. The choice does not affect completeness or the topology of the space. [/remark]