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Mazón, J. M.; Rossi, J. D.; Toledo, J.

On the best Lipschitz extension problem for a discrete distance and the discrete \(\infty \)-Laplacian. (English. French summary) Zbl 1231.49003

J. Math. Pures Appl. (9) 97, No. 2, 98-119 (2012).
Summary: This paper concerns the best Lipschitz extension problem for a discrete distance that counts the number of steps. We relate this absolutely minimizing Lipschitz extension with a discrete \(\infty \)-Laplacian problem, which arises as the dynamic programming formula for the value function of some \(\epsilon \)-tug-of-war games. As in the classical case, we obtain the absolutely minimizing Lipschitz extension of a datum \(f\) by taking the limit as \(p\to \infty \) in a nonlocal \(p\)-Laplacian problem.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
49J45 Methods involving semicontinuity and convergence; relaxation
45G10 Other nonlinear integral equations

Keywords:

Lipschitz extension; nonlocal \(p\)-Laplacian problem; infinity Laplacian; tug-of-war games
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References:

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