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A limiting free boundary problem with gradient constraint and Tug-of-War games

  • P. Blanc
  • J. V. da Silva
  • J. D. RossiEmail author
Article
  • 14 Downloads

Abstract

In this manuscript we deal with regularity issues and the asymptotic behaviour (as \(p \rightarrow \infty \)) of solutions for elliptic free boundary problems of \(p-\)Laplacian type (\(2 \le p< \infty \)):
$$\begin{aligned} -\Delta _p u(x) + \lambda _0(x)\chi _{\{u>0\}}(x) = 0 \quad \text{ in } \quad \Omega \subset {\mathbb {R}}^N, \end{aligned}$$
with a prescribed Dirichlet boundary data, where \(\lambda _0>0\) is a bounded function and \(\Omega \) is a regular domain. First, we prove the convergence as \(p\rightarrow \infty \) of any family of solutions \((u_p)_{p\ge 2}\), as well as we obtain the corresponding limit operator (in non-divergence form) ruling the limit equation,
$$\begin{aligned} \left\{ \begin{array}{lllll} \max \left\{ -\Delta _{\infty } u_{\infty }, \,\, -|\nabla u_{\infty }| + \chi _{\{u_{\infty }>0\}}\right\} &{} = &{} 0 &{} \text{ in } &{} \Omega \cap \{u_{\infty } \ge 0\} \\ u_{\infty } &{} = &{} F &{} \text{ on } &{} \partial \Omega . \end{array} \right. \end{aligned}$$
Next, we obtain uniqueness for solutions to this limit problem. Finally, we show that any solution to the limit operator is a limit of value functions for a specific Tug-of-War game.

Keywords

Lipschitz regularity estimates Free boundary problems \(\infty \)-Laplace operator Existence/uniqueness of solutions Tug-of-War games 

Mathematics Subject Classification

35J92 35D40 91A80 

Notes

Acknowledgements

This work was partially supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET-Argentina).

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.FCEyN, Department of MathematicsUniversidad de Buenos AiresBuenos AiresArgentina

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