Existence of Solutions in the Trace Sense

In previous Chapters, we introduced two different notions of solutions to the least gradient problem. In Chap. 1, we understood the boundary condition in the trace sense, i.e. we required that the trace of the solution equals the boundary datum. Then, the Sternberg-Williams-Ziemer construction guarantees existence of a solution, provided that \(\Omega \subset \mathbb {R}^N\) (with \(N \ge 2\) ) is strictly convex and the boundary datum is continuous. On the other hand, in Chap. 2, in the definition of weak solutions we understood the Dirichlet boundary condition in a weaker sense; this enabled us to prove existence of solutions for an arbitrary Lipschitz domain and integrable boundary datum. Our goal in this Chapter is to bridge a gap between these two approaches. We introduce a notion of strong solutions (in the trace sense) to the Dirichlet problem for the anisotropic 1-Laplace operator and present several approaches to existence of solutions in the strong sense for discontinuous boundary data.