Mather measures selected by an approximation scheme
Authors:
Diogo Gomes, Renato Iturriaga, Héctor SánchezMorgado and Yifeng Yu
Journal:
Proc. Amer. Math. Soc. 138 (2010), 35913601
MSC (2010):
Primary 37J20, 35J70, 37J50
Published electronically:
April 27, 2010
MathSciNet review:
2661558
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References 
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Additional Information
Abstract: In this note, we will identify Mather measures selected by Evans's variational approach in 1d. Motivated by the low dimension case, we conjecture that Evans's approximation scheme might catch the whole Mather set in all dimensions. We also discuss the connection with another approximation scheme in the works of Anantharaman, Evans and Gomes.
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 N. Anantharaman, On the zerotemperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics. J. Eur. Math. Soc. 6 (2004), no. 2, 207276. MR 2055035 (2005i:82004)
 [AIPS]
 N. Anantharaman, R. Iturriaga, P. Padilla, H. SánchezMorgado, Physical solutions of the HamiltonJacobi equation. Discrete Contin. Dyn. Syst. Ser. B 5 (2005), no. 3, 513528. MR 2151719 (2006e:35031)
 [Ar1]
 G. Aronsson, Minimization problems for the functional . Ark. Mat. 6 (1965), 3353. MR 0196551 (33:4738)
 [Ar2]
 G. Aronsson, Minimization problems for the functional . II. Ark. Mat. 6 (1966), 409431. MR 0203541 (34:3391)
 [Ar3]
 G. Aronsson, Minimization problems for the functional . III. Ark. Mat. 7 (1969), 509512 (1969). MR 0240690 (39:2035)
 [B]
 U. Bessi, AubryMather theory and HamiltonJacobi equations. Comm. Math. Phys. 235 (2003), no. 3, 495511. MR 1974512 (2004e:37096)
 [C]
 G. Contreras, R. Iturriaga, G. P. Paternain, M. Paternain, Lagrangian graphs, minimizing measures and Mañé's critical values. Geom. Funct. Anal. 8 (1998), no. 5, 788809. MR 1650090 (99f:58075)
 [E1]
 L. C. Evans, Some new PDE methods for weak KAM theory. Calc. Var. Partial Differential Equations 17 (2003), no. 2, 159177. MR 1986317 (2004e:37097)
 [E2]
 L. C. Evans, Towards a quantum analog of weak KAM theory. Comm. Math. Phys. 244 (2004), no. 2, 311334. MR 2031033 (2005c:81063)
 [F]
 A. Fathi, Weak KAM theorem in Lagrangian dynamics, Cambridge University Press, in press.
 [G1]
 D. Gomes, A stochastic analogue of AubryMather theory. Nonlinearity 15 (2002), no. 3, 581603. MR 1901094 (2003b:37096)
 [G2]
 D. Gomes, Generalized Mather problem and selection principles for viscosity solutions and Mather measures. Adv. Calc. Var. 1 (2008), no. 3, 291307. MR 2458239
 [JKM]
 H. R. Jauslin, H. O. Kreiss, J. Moser, On the forced Burgers equation with periodic boundary conditions. Differential Equations: La Pietra 1996. Proc. Symp. Pure Math. 65, Amer. Math. Soc., Providence, RI, 1999. MR 1662751 (99m:35208)
 [LPV]
 P. L. Lions, G. Papanicolaou, S. R. S. Varadhan, Homogenization of HamiltonJacobi equations, unpublished manuscript, circa 1988.
 [M]
 J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z. 207 (1991), no. 2, 169207. MR 1109661 (92m:58048)
 [Y1]
 Y. Yu, variational problems, Aronsson equations and weak KAM theory, Ph.D. dissertation, University of California, Berkeley, 2005.
 [Y2]
 Y. Yu, A remark on the semiclassical measure from with a degenerate potential . Proc. Amer. Math. Soc. 135 (2007), no. 5, 14491454. MR 2276654 (2007k:35079)
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Additional Information
Diogo Gomes
Affiliation:
Departamento de Matemática and CAMGSD, Instituto Superior Técnico, Lisboa, Portugal
Email:
dgomes@math.ist.utl.pt
Renato Iturriaga
Affiliation:
Centro de Investigación en Matemáticas, Guanajuato, México
Email:
renato@cimat.mx
Héctor SánchezMorgado
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, México DF 04510, México
Email:
hector@matem.unam.mx
Yifeng Yu
Affiliation:
Department of mathematics, University of California at Irvine, Irvine, California 92697
Email:
yyu1@math.uci.edu
DOI:
http://dx.doi.org/10.1090/S000299391010361X
PII:
S 00029939(10)10361X
Received by editor(s):
September 29, 2009
Received by editor(s) in revised form:
December 31, 2009
Published electronically:
April 27, 2010
Additional Notes:
The first author was partially supported by the CAMGSD/IST through the FCT Program POCTI/FEDER and by grants DENO/FCTPT (PTDC/EEAACR/67020/2006 and UTAustin/MAT/0057/2008
The second author was partially supported by Conacyt grant 83739
The fourth author was partially supported by NSF grants D0848378 and D0901460
Communicated by:
Yingfei Yi
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
