A limiting free boundary problem ruled by Aronsson’s equation

Rossi, Julio; Teixeira, Eduardo

doi:10.1090/S0002-9947-2011-05322-5

A limiting free boundary problem ruled by Aronsson’s equation
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by Julio D. Rossi and Eduardo V. Teixeira

Trans. Amer. Math. Soc. 364 (2012), 703-719

DOI: https://doi.org/10.1090/S0002-9947-2011-05322-5

Published electronically: September 13, 2011

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Abstract:

We study the behavior of a $p$-Dirichlet optimal design problem with volume constraint for $p$ large. As the limit of $p$ goes to infinity, we find a limiting free boundary problem governed by the infinity-Laplacian operator. We find a necessary and sufficient condition for uniqueness of the limiting problem and, under such a condition, we determine precisely the optimal configuration for the limiting problem. Finally, we establish convergence results for the free boundaries.

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