An optimal transportation problem related to the limits of solutions of local and nonlocal $$p$$ p -Laplace-type problems

Abstract

We investigate the asymptotic behavior of minimizers of problems related to the \(p\)-Laplace equation \(-\Delta _p u=f\). As \(p\rightarrow \infty \), the minimizers converge (up to a subsequence) to a function, which maximizes the functional \(I(u)= \int fu\) with appropriate constraints. The main result of this paper is that the problem of maximizing \(I\) with Dirichlet boundary condition can be identified as a dual problem of a certain mass transportation problem. Our approach applies to the limits of both \(p\)-Laplace type problems in classical Sobolev spaces and analogous nonlocal problems in fractional Sobolev spaces.