Existence and Characterization of Weak Solutions

In the previous Chapter, we studied the least gradient problem for continuous boundary data \(f \in C(\partial \Omega )\) . Then, the construction by Sternberg, Williams and Ziemer provided us with a unique solution under some geometric assumptions on \(\Omega \) . This time, we only assume that \(\Omega \) is a bounded Lipschitz domain in \(\mathbb R^N\) and allow for any \(f \in L^1(\partial \Omega )\) . We will identify the Euler–Lagrange equation for the least gradient problem, which is the Dirichlet problem for the 1-Laplacian ( \(-\textrm{div} (\frac{Du}{|Du|})=0\) ). We then prove existence of weak solutions using an approximation by a sequence of p-harmonic functions and establish an equivalence between functions of least gradient and weak solutions. One of the key differences with respect to the previous Chapter is that weak solutions do not necessarily satisfy the boundary condition in the trace sense; this is visible both in relation to the shape of the domain and regularity of boundary data. We also present several examples highlighting main properties of weak solutions.