Approximation by Nonlocal Problems

In this Chapter we study a nonlocal counterpart of the least gradient problem, i.e. the minimization of the functional \(\begin{aligned} \mathcal {J}_{\psi } (u):= \frac{1}{2}\int _{\Omega _J}\int _{\Omega _J} J(x-y) | u_{\psi }(y) - u_{\psi }(x) | \, dx \, dy \end{aligned}\) defined over \(L^1(\Omega )\) and corresponding to the Dirichlet problem for the nonlocal 1-Laplacian operator. Our approach is similar to Chap. : we introduce a notion of variational solutions, i.e. find an Euler-Lagrange type characterization of minimizers of \(\mathcal {J}_\psi \) , and study their properties. This is achieved using an approximation by solutions to a nonlocal p-Laplacian equation for \(p > 1\) . We also discuss the relationship between the variational solutions and a median value property. In the last part of this Chapter, we examine a link between the local and nonlocal Dirichlet problems for the 1-Laplacian.