Navegar

Colección

Datos Estadísticas

Artículo

Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

In this paper, we study the behavior as p→ ∞ of eigenvalues and eigenfunctions of a system of p-Laplacians, that is (Formula presented.) in a bounded smooth domain Ω. Here α+ β= p. We assume that α/p→Γ and β/p→1-Γ as p→ ∞ and we prove that for the first eigenvalue λ1,p we have (λ1,p)1/p→λ∞=1/maxx∈Ωdist(x,∂Ω).Concerning the eigenfunctions (up, vp) associated with λ1,p normalized by ∫Ω|up|α|vp|β=1, there is a uniform limit (u∞, v∞) that is a solution to a limit minimization problem as well as a viscosity solution to (Formula presented.) In addition, we also analyze the limit PDE when we consider higher eigenvalues. © 2015, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg.

Registro:

Documento: Artículo
Título:The limit as p→ ∞ in the eigenvalue problem for a system of p-Laplacians
Autor:Bonheure, D.; Rossi, J.D.; Saintier, N.
Filiación:Département de Mathématique, Université libre de Bruxelles, CP 214, Boulevard du Triomphe, Brussels, 1050, Belgium
Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Ciudad Universitaria, Pab 1 (1428), Buenos Aires, Argentina
Palabras clave:Infinity Laplacian; Nonlinear eigenvalue problem; p-Laplacian; Viscosity solutions
Año:2016
Volumen:195
Número:5
Página de inicio:1771
Página de fin:1785
DOI: http://dx.doi.org/10.1007/s10231-015-0547-2
Título revista:Annali di Matematica Pura ed Applicata
Título revista abreviado:Ann. Mat. Pura Appl.
ISSN:03733114
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03733114_v195_n5_p1771_Bonheure

Referencias:

  • 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
  • Bhattacharya, T., DiBenedetto, E., Manfredi, J.J., Limits as p→ ∞ of Δ pup= f and related extremal problems (1989) Rend. Sem. Mat. Univ. Politec. Torino, 1991, pp. 15-68
  • Boccardo, L., de Figueiredo, D.G., Some remarks on a system of quasilinear elliptic equations (2002) Nonlinear Differ. Equ. Appl, 9, pp. 309-323
  • Caselles, V., Morel, J.M., Sbert, C., An axiomatic approach to image interpolation (1998) IEEE Trans. Image Process, 7, pp. 376-386
  • Champion, T., De Pascale, L., Jimenez, C., The ∞ -eigenvalue problem and a problem of optimal transportation (2009) Commun. Appl. Anal, 13 (4), pp. 547-565
  • 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
  • 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, 66, pp. 349-366
  • Garcia-Azorero, J., Manfredi, J.J., Peral, I., Rossi, J.D., Steklov eigenvalue for the ∞ -Laplacian (2006) Rendiconti Lincei, 17 (3), pp. 199-210
  • Fleckinger, J., Mansevich, R.F., Stavrakakis, N.M., de Thlin, F., Principal eigenvalues for some quasilinear elliptic equations on Rn (1997) Adv. Differ. Equ, 2 (6), pp. 981-1003
  • Hynd, R., Smart, C.K., Yu, Y., Nonuniqueness of infinity ground states (2013) Calc. Var. PDE, 48 (3), pp. 545-554
  • Juutinen, P., Lindqvist, P., On the higher eigenvalues for the ∞ - eigenvalue problem (2005) Calc. Var. Partial Differ. Equ, 23 (2), pp. 169-192
  • Juutinen, P., Lindqvist, P., Manfredi, J.J., The ∞ -eigenvalue problem (1999) Arch. Ration. Mech. Anal, 148, pp. 89-105
  • Juutinen, P., Lindqvist, P., Manfredi, J.J., On the equivalence of viscosity solutions and weak solutions for a quasilinear equation (2001) SIAM J. Math. Anal, 33 (3), pp. 699-717
  • Manfredi, J.J., Rossi, J.D., Urbano, J.M., p(x) Inst Henri Poincaré (2009) C. Anal. Non Linéaire, 26 (6), pp. 2581-2595
  • de Napoli, P.L., Pinasco, J.P., Estimates for eigenvalues of quasilinear elliptic systems (2006) J. Differ. Equ, 227, pp. 102-115
  • Peres, Y., Schramm, O., Sheffield, S., Wilson, D.B., Tug-of-war and the infinity Laplacian (2009) J. Am. Math. Soc, 22, pp. 167-210
  • Peres, Y., Sheffield, S., Tug-of-war with noise: a game theoretic view of the p -Laplacian (2008) Duke Math. J, 145, pp. 91-120
  • Houston J. Math, , Rossi, J.D., Saintier, N.: On the first nontrivial eigenvalue of the ∞ -Laplacian with Neumann boundary conditions. (to appear)
  • Rossi, J.D., Saintier, N., The limit as p→ + ∞ of the first eigenvalue for the p -Laplacian with mixed Dirichlet and Robin boundary conditions (2015) Nonlinear Anal, 119, pp. 167-178
  • Villani, C., (2009) Optimal transport, old and new, Grundlehren der Mathematischen Wissenschaften, 338, , Springer, Berlin
  • Yu, Y., Some properties of the ground sates of the infinity Laplacian (2007) Indiana Univ. Math. J, 56 (2), pp. 947-964
  • Zographopoulos, N., p -Laplacian systems at resonance (2004) Appl. Anal, 83 (5), pp. 509-519

Citas:

---------- APA ----------
Bonheure, D., Rossi, J.D. & Saintier, N. (2016) . The limit as p→ ∞ in the eigenvalue problem for a system of p-Laplacians. Annali di Matematica Pura ed Applicata, 195(5), 1771-1785.
http://dx.doi.org/10.1007/s10231-015-0547-2
---------- CHICAGO ----------
Bonheure, D., Rossi, J.D., Saintier, N. "The limit as p→ ∞ in the eigenvalue problem for a system of p-Laplacians" . Annali di Matematica Pura ed Applicata 195, no. 5 (2016) : 1771-1785.
http://dx.doi.org/10.1007/s10231-015-0547-2
---------- MLA ----------
Bonheure, D., Rossi, J.D., Saintier, N. "The limit as p→ ∞ in the eigenvalue problem for a system of p-Laplacians" . Annali di Matematica Pura ed Applicata, vol. 195, no. 5, 2016, pp. 1771-1785.
http://dx.doi.org/10.1007/s10231-015-0547-2
---------- VANCOUVER ----------
Bonheure, D., Rossi, J.D., Saintier, N. The limit as p→ ∞ in the eigenvalue problem for a system of p-Laplacians. Ann. Mat. Pura Appl. 2016;195(5):1771-1785.
http://dx.doi.org/10.1007/s10231-015-0547-2