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.
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Acknowledgments
The author would like to thank Petri Juutinen for introducing the author to the problems discussed in this paper and Julio Rossi for sharing his ideas about how to generalize the results. The author would also like to thank the reviewers for their helpful comments. The author has been supported by University of Jyväskylä and Väisälä Foundation.
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Jylhä, H. An optimal transportation problem related to the limits of solutions of local and nonlocal \(p\)-Laplace-type problems. Rev Mat Complut 28, 85–121 (2015). https://doi.org/10.1007/s13163-014-0147-5
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DOI: https://doi.org/10.1007/s13163-014-0147-5