The inhomogeneous $p$-Laplacian equation with Neumann boundary conditions in the limit $p\to \mathrm{\infty }$

We investigate the limiting behavior of solutions to the inhomogeneous $p$-Laplacian equation $-{\mathrm{\Delta }}_{p}u={\mu }_p$ subject to Neumann boundary conditions. For right hand sides which are arbitrary signed measures we show that solutions converge to a Kantorovich potential associated with the geodesic Wasserstein-$1$ distance. In the regular case with continuous right hand sides we characterize the limit as viscosity solution to an infinity Laplacian / eikonal type equation.