Lower semicontinuity and existence results for anisotropic TV functionals with signed measure data

TL;DR

The paper develops a rigorous variational framework for anisotropic total variation functionals with signed measure data on BV, under Dirichlet boundary conditions. Central to the results is a sharp anisotropic isoperimetric condition (IC) that governs lower semicontinuity, coercivity, and relaxation, enabling existence of minimizers via the direct method and ensuring recovery sequences from the Dirichlet class. The authors derive multiple equivalent IC formulations, demonstrate sharpness through detailed examples, and extend semicontinuity to a refined parametric setting. A key contribution is the identification of signed ICs that permit interaction between the BV-variation term and the measure data, along with a recovery framework that links BV minimizers to W^{1,1}_{u_0} minimizers. The work provides a comprehensive structure for anisotropic BV problems with signed measures, including an explicit coercivity criterion, relaxation results, and conditions for the consistency of infima across BV and Dirichlet frameworks, with potential implications for image processing models and variational PDEs with measure data.

Abstract

We study the minimization of anisotropic total variation functionals with additional measure terms among functions of bounded variation subject to a Dirichlet boundary condition. More specifically, we identify and characterize certain isoperimetric conditions, which prove to be sharp assumptions on the signed measure data in connection with semicontinuity, existence, and relaxation results. Furthermore, we present a variety of examples which elucidate our assumptions and results.

Figures (4)

• Figure 1: The domain $\Omega$ of Example \ref{['exp:non-exist']} and the graph of the potential minimizer of $\widehat{\Phi}$ (red surface) with some level sets (colored arcs and pink disc) and with the position of the pole (pink dot).
• Figure 2: The iterates $\Delta_0$, $\Delta_1$, $\Delta_2$, and $\Delta_3$ of Example \ref{['exp:til2']}
• Figure 3: The portions of $C_i$ and $H$ (thick green and blue lines) covered by $A^+$ (orange area) are estimated by integrating along the green and blue stripes, respectively, till they leave $A^+$. This works out even in the blue intersection area, since ${\mathcal{H}}^{N-1}$-a.e. point where blue stripes leave $A^+$ in both right and upward direction is in particular contained in the upward red-colored part of $\partial^\ast\!A$ which contributes to $\mathrm{P}_\varphi(A)$ with its larger $1$-norm length.
• Figure 4: An illustration of $\sigma_k$ for $k\in\{1,2\}$, where $\sigma_k\equiv0$ on $\Delta_k$ (orange regions), $\sigma_k\in{[0,1]}^2\cap\partial\mathrm{B}_1$ on blue-arrow regions, $\sigma_k\equiv({-}1,{-}1)$ on red-arrow regions, and $\sigma_k\in{[{-}1,0]}^2$ in red-blue-superposition regions.

Theorems & Definitions (115)

• Theorem 2.1: coarea formula for Lipschitz functions
• Remark 2.2
• Remark 2.3
• proof
• Lemma 2.4
• Theorem 2.5: pointwise convergence of $\mathrm{BV}$ functions
• Theorem 2.6: De Giorgi's structure theorem; partial statement
• Theorem 2.7: pasting $\mathrm{BV}$ functions across reduced boundaries
• Theorem 2.8: continuity of the boundary trace operator
• Theorem 2.9: isoperimetric inequality