Total variation seminorm on a domain ($\mathrm{TV}(u)$)

For $u\in L^1(\Omega )$, the total variation seminorm isIt extends the integral of the magnitude of the gradient to nonsmooth functions.

A function $u\in L^1(\Omega )$ has bounded variation when $\mathrm{TV}(u)<\infty$. The space $BV(\Omega )$ is a Banach space with norm $\Vert u{\Vert }_{L^1}+\mathrm{TV}(u)$.

The zero-mean bounded-variation space isRemoving the constant nullspace makes total variation coercive on this subspace.

On a bounded Lipschitz domain,It reduces to $\Vert u{\Vert }_{L^1}\le C_{\Omega }\mathrm{TV}(u)$ on $B{V}_0(\Omega )$.