Bounded variation and distributional gradients

Let Omega be open and let u be integrable there. We say u belongs to BV when its distributional gradient is a finite vector-valued Radon measure. Integration by parts defines that measure using compactly supported smooth vector fields, so no boundary trace of u is inserted at the edge of Omega. The total variation measure records the size of the vector measure without cancelling opposite directions. Its mass is the BV seminorm.

Equivalently, total variation is the supremum of the divergence pairing over compactly supported test fields with pointwise norm at most one. A smooth function has gradient measure equal to its ordinary gradient times Lebesgue measure, and its variation is the integral of gradient magnitude. General BV functions may also have singular gradient components. The definition concerns an L1 equivalence class, so changing a value on a null set does not change either its derivative measure or its variation.

In one dimension a jump from a left value to a right value contributes their signed difference as a Dirac mass. The variation uses its absolute magnitude, so a rise and a later fall both count. A function with zero classical derivative off finitely many jumps may therefore have positive variation. A Sobolev W1,1 function instead has an absolutely continuous distributional gradient; jump masses exclude membership in that class on an interval containing the jumps.

Not every singular BV derivative is a jump. The Cantor function gives a continuous example whose derivative is singular and diffuse, illustrating why a jump-only description is incomplete. Fine BV theory uses approximate limits and one-sided traces rather than arbitrary values written on a jump surface. Those representative conventions matter when composing or tracing functions, but they do not change the underlying distributional derivative of an almost-everywhere class.

The BV coarea identity expresses variation as the integral of the relative perimeters of superlevel sets. For almost every level, these sets have finite perimeter. This is distinct from the Lipschitz coarea formula: here the gradient is a measure and the perimeter integrand can detect jumps. Values assigned on a null set do not change superlevel sets up to null sets, consistent with the equivalence-class convention.

Lower semicontinuity says that an L1-convergent sequence cannot produce more variation in its limit than its limiting lower variation bound. On bounded Lipschitz domains, uniformly bounded L1 norms and variations give L1 subsequential compactness. Local forms use compact containment and local bounds. When extending a BV function by zero beyond its domain, boundary trace jumps may add variation, so relative and full-space compactness estimates should not be interchanged without checking the extension.