Duality Approach

In this Chapter, we again want to study the Euler-Lagrange equations related to the anisotropic least gradient problem \(\begin{aligned} \min \bigg \{ \int _\Omega |Du|_\phi : \quad u \in BV(\Omega ), \quad u|_{\partial \Omega } = f \bigg \}, \end{aligned}\) but from a different perspective. In Chaps. and (respectively in the isotropic and anisotropic case), we relied on approximations; in Chap. , this was approximation by solutions to the p-Laplace equation as \(p \rightarrow 1\) , and in Chap. it was the Moreau-Yosida approximation. Here, we use a different (and perhaps simpler) approach. Instead of solving a sequence of approximate problems, we will use the duality theory from convex analysis. In the second half of this Chapter, we apply this technique to a variant of the least gradient problem where we prescribe the boundary datum only on a relatively open subset \(\Gamma \) of the boundary. We also present some examples showing that the behavior of solutions can be qualitatively different to the case when the boundary datum is prescribed on the full boundary.