On the solutions of a double-phase Dirichlet problem involving the 1-Laplacian
Alexandros Matsoukas, Nikos Yannakakis
TL;DR
The paper analyzes a double-phase Dirichlet problem driven by the 1-Laplacian with non-homogeneous boundary data $u=h$. The authors approximate by $p$-phase problems for $p>1$ and employ a BV/Anzellotti framework, obtaining existence and uniqueness of a weak solution and a variational minimization description. They show that, as $p\to1$, a subsequence converges to a limit $u$ with an associated bounded vector field $z$ satisfying a weak formulation and the condition $z\cdot\nabla u=|\nabla u|$, thus solving the limiting problem. A further result provides a precise variational characterization: the limit is the unique minimizer of $\mathcal{I}(u)=\int_\Omega|\nabla u|\,dx+\frac{1}{q}\int_\Omega a(x)|\nabla u|^q\,dx$, linking the PDE with a convex functional in a weighted, BV-compatible setting. These contributions extend the 1-Laplacian theory to double-phase settings and establish a rigorous p-to-1 transition for anisotropic materials modeled by the double-phase functional.
Abstract
In this paper we study a double-phase problem involving the 1-Laplacian with non-homogeneous Dirichlet boundary conditions and show the existence and uniqueness of a solution in a suitable weak sense. We also provide a variational characterization of this solution via the corresponding minimization problem.