The coarea formula represents a profound generalization of Fubini’s theorem, offering a powerful framework for integrating functions over level sets of a Lipschitz map. At its core, it provides a way to decompose the integral of a function over a domain in ℝⁿ into an integral over the level sets of a function mapping to ℝᵏ, typically with k less than n. This decomposition reveals how the mass of a function distributes across fibers of a mapping, making it an indispensable tool in geometric measure theory and partial differential equations.
Foundational Concepts and Intuition
To grasp the coarea formula, one must first consider the familiar slicing principle in lower dimensions. Imagine calculating the area of a curved surface in three-dimensional space. One effective strategy is to project the surface onto a plane and then integrate the lengths of the intersection curves, or level sets, created by slicing the surface with planes parallel to the projection axis. The coarea formula elevates this intuitive slicing process to a high-dimensional Riemannian setting, where the "slices" are generalized to level sets of a Sobolev function. The formula precisely quantifies the relationship between the integral over the original space and the integrals over these nested, lower-dimensional manifolds.
The Mathematical Statement
Formally, let Ω be a Lipschitz domain in ℝⁿ, and let u: Ω → ℝᵏ be a Lipschitz map with k ≤ n. The coarea formula states that for any integrable function g on Ω, the following equality holds:
∫ Ω g(x) J u (x) dx = ∫ ℝᵏ ( ∫ u⁻¹(t) g(x) dH n-k (x) ) dt
In this expression, J u (x) denotes the Jacobian of the mapping u, specifically the square root of the determinant of the matrix product of the gradient of u with its transpose. The term dH n-k represents the (n-k)-dimensional Hausdorff measure, which acts as the integration element over the (n-k)-dimensional level set u⁻¹(t). This elegant equation asserts that the weighted integral of the Jacobian over the domain is equal to the integral of the Hausdorff measures of the level sets, weighted by the integral of the function along those sets.
The Role of the Jacobian
The Jacobian factor J u (x) is critical, as it accounts for the distortion of volume when the mapping u compresses or stretches space. It effectively weights the contribution of each point x in the domain Ω. When u represents a height function, for instance, the Jacobian measures the local area expansion factor of the projection onto the height plane. Without this factor, the formula would fail to conserve the total mass of the integral during the decomposition process.
Connections to the Coarea Inequality
Closely related to the coarea formula is the coarea inequality, which provides a lower bound for the integral of the Jacobian. This inequality is particularly useful in analysis, as it connects the total variation of a function to the Hausdorff measure of its level sets. For a Lipschitz function u: ℝⁿ → ℝᵏ, the coarea inequality asserts that for any non-negative integrable function g, the integral of g times the Jacobian is bounded below by the integral over t of the integral over the level set with respect to the (n-k)-dimensional measure. This inequality is fundamental in proving regularity results for functions of bounded variation and in understanding the structure of singularities.
