The authors study as \(p\rightarrow \infty\), in a convex smooth domain \(\Omega\), the boundary problem involving a \(p\)-Laplacian \(-\text{div}(|Du|^{p-2}Du)=f\), where supp\((f)\subset \Omega\) and with the mixed boundary conditions \(u=0\) on \(\Gamma\) and \(|Du|^{p-2}\frac{\partial u}{\partial v}=0\) on \(\Omega \backslash \Gamma \). \(\Gamma \) is a smooth submanifold of \(\partial \Omega\) (the window) satisfying some further conditions. The limit problem as \(p\rightarrow \infty\) is considered in variational formulation, next in the weak formulation and finally in the viscosity sense.
The authors connect the problem under consideration to the optimal mass transport problem. Namely, the most efficient way of transport \(f(x)dx\) to the window \(\Gamma\) with linear cost is determined.