A mixed problem for the infinity Laplacian via Tug-of-War games

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In this paper we prove that a function uC} overlineΩ is the continuous value of the Tug-of-War game described in Y. Peres et al. (J. Am. Math. Soc., 2008, to appear) if and only if it is the unique viscosity solution to the infinity Laplacian with mixed boundary conditions equation presented. By usi...

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Autor principal: Charro, F.
Otros Autores: García Azorero, J., Rossi, J.D
Formato: Capítulo de libro
Lenguaje:Inglés
Publicado: 2009
Acceso en línea:Registro en Scopus
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100 1 |a Charro, F. 
245 1 2 |a A mixed problem for the infinity Laplacian via Tug-of-War games 
260 |c 2009 
270 1 0 |m Charro, F.; Departamento de Matemáticas, U. Autonoma de Madrid, Madrid 28049, Spain; email: fernando.charro@uam.es 
506 |2 openaire  |e Política editorial 
504 |a Ambrosio, L., Lecture Notes on Optimal Transport Problems, , CVGMT preprint server 
504 |a Aronsson, G., Extensions of functions satisfying Lipschitz conditions (1967) Ark. Mat., 6, pp. 551-561 
504 |a Aronsson, G., Crandall, M.G., Juutinen, P., A tour of the theory of absolutely minimizing functions (2004) Bull. Am. Math. Soc., 41, pp. 439-505 
504 |a Barles, G., Fully nonlinear Neumann type conditions for second-order elliptic and parabolic equations (1993) J. Differ. Equ., 106, pp. 90-106 
504 |a Barles, G., Busca, J., Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order terms (2001) Comm. Partial Diff. Equ., 26, pp. 2323-2337 
504 |a Barron, E.N., Evans, L.C., Jensen, R., The infinity Laplacian, Aronsson's equation and their generalizations (2008) Trans. Am. Math. Soc., 360, pp. 77-101 
504 |a Bhattacharya, T., Di Benedetto, E., Manfredi, J., Limits as p → ∞ of Δ p u p = f and related extremal problems (1991) Rend. Sem. Mat. Univ. Politec., Torino, pp. 15-68 
504 |a Crandall, M.G., Gunnarsson, G., Wang, P., Uniqueness of ∞-harmonic functions and the eikonal equation (2007) Comm. Partial Diff. Equ., 32, pp. 1587-1615 
504 |a Crandall, M.G., Ishii, H., Lions, P.L., User's guide to viscosity solutions of second order partial differential equations (1992) Bull. Am. Math. Soc., 27, pp. 1-67 
504 |a Evans, L.C., Gangbo, W., Differential equations methods for the Monge-Kantorovich mass transfer problem (1999) Mem. Am. Math. Soc., 137, p. 653 
504 |a García-Azorero, J., Manfredi, J.J., Peral, I., Rossi, J.D., The Neumann problem for the ∞-Laplacian and the Monge-Kantorovich mass transfer problem (2007) Nonlinear Anal. T.M.A., 66, pp. 349-366. , 2 
504 |a Jensen, R., Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient (1993) Arch. Rational Mech. Anal., 123, pp. 51-74 
504 |a Juutinen, P., Principal eigenvalue of a badly degenerate operator and applications (2007) J. Differ. Equ., 236, pp. 532-550 
504 |a Juutinen, P., Lindqvist, P., Manfredi, J., (2001) The Infinity Laplacian: Examples and Observations, Papers on Analysis, pp. 207-217. , Rep. Univ. Jyväskylä Dep. Math. Stat., 83, Univ. Jyväskylä, Jyväskylä 
504 |a Kohn, R.V., Serfaty, S., A deterministic-control-based approach to motion by curvature (2006) Commun. Pure Appl. Math., 59, pp. 344-407. , 3 
504 |a Sthocastic games and applications (2003) NATO Science Series, pp. 27-36. , Neymann, A., Sorin, S. (eds.) 
504 |a Oberman, A.M., A convergent difference scheme for the infinity-Laplacian: Construction of absolutely minimizing Lipschitz extensions (2005) Math. Comput., 74, pp. 1217-1230 
504 |a Peres, Y., Sheffield, S., Tug-of-war with Noise: A Game Theoretic View of the P-Laplacian, , preprint 
504 |a Peres, Y., Schramm, O., Sheffield, S., Wilson, D., Tug-of-war and the infinity Laplacian (2008) J. Am. Math. Soc., , To appear in 
520 3 |a In this paper we prove that a function uC} overlineΩ is the continuous value of the Tug-of-War game described in Y. Peres et al. (J. Am. Math. Soc., 2008, to appear) if and only if it is the unique viscosity solution to the infinity Laplacian with mixed boundary conditions equation presented. By using the results in Y. Peres et al. (J. Am. Math. Soc., 2008, to appear), it follows that this viscous PDE problem has a unique solution, which is the unique absolutely minimizing Lipschitz extension to the whole Ω (in the sense of Aronsson (Ark. Mat. 6:551-561, 1967) and Y. Peres et al. (J. Am. Math. Soc., 2008, to appear)) of the Lipschitz boundary data F:ΓD R . © 2008 Springer-Verlag.  |l eng 
593 |a Departamento de Matemáticas, U. Autonoma de Madrid, Madrid 28049, Spain 
593 |a Departamento de Matemática, FCEyN, Ciudad Universitaria, Pab I, (1428) Buenos Aires, Argentina 
700 1 |a García Azorero, J. 
700 1 |a Rossi, J.D. 
773 0 |d 2009  |g v. 34  |h pp. 307-320  |k n. 3  |p Calc. Var. Partial Differ. Equ.  |x 09442669  |t Calculus of Variations and Partial Differential Equations 
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856 4 0 |u https://doi.org/10.1007/s00526-008-0185-2  |y DOI 
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