Mather measures selected by an approximation scheme

Authors:
Diogo Gomes, Renato Iturriaga, Héctor Sánchez-Morgado and Yifeng Yu

Journal:
Proc. Amer. Math. Soc. **138** (2010), 3591-3601

MSC (2010):
Primary 37J20, 35J70, 37J50

Published electronically:
April 27, 2010

MathSciNet review:
2661558

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Abstract | References | Similar Articles | Additional Information

Abstract: In this note, we will identify Mather measures selected by Evans's variational approach in 1-d. 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.

**[A]**Nalini Anantharaman,*On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics*, J. Eur. Math. Soc. (JEMS)**6**(2004), no. 2, 207–276. MR**2055035****[AIPS]**Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, and Héctor Sánchez-Morgado,*Physical solutions of the Hamilton-Jacobi equation*, Discrete Contin. Dyn. Syst. Ser. B**5**(2005), no. 3, 513–528. MR**2151719**, 10.3934/dcdsb.2005.5.513**[Ar1]**Gunnar Aronsson,*Minimization problems for the functional 𝑠𝑢𝑝ₓ𝐹(𝑥,𝑓(𝑥),𝑓′(𝑥))*, Ark. Mat.**6**(1965), 33–53 (1965). MR**0196551****[Ar2]**Gunnar Aronsson,*Minimization problems for the functional 𝑠𝑢𝑝ₓ𝐹(𝑥,𝑓(𝑥),𝑓′(𝑥)). II*, Ark. Mat.**6**(1966), 409–431 (1966). MR**0203541****[Ar3]**Gunnar Aronsson,*Minimization problems for the functional 𝑠𝑢𝑝ₓ𝐹(𝑥,𝑓(𝑥),𝑓′(𝑥)). III*, Ark. Mat.**7**(1969), 509–512. MR**0240690****[B]**Ugo Bessi,*Aubry-Mather theory and Hamilton-Jacobi equations*, Comm. Math. Phys.**235**(2003), no. 3, 495–511. MR**1974512**, 10.1007/s00220-002-0781-5**[C]**G. Contreras, R. Iturriaga, G. P. Paternain, and M. Paternain,*Lagrangian graphs, minimizing measures and Mañé’s critical values*, Geom. Funct. Anal.**8**(1998), no. 5, 788–809. MR**1650090**, 10.1007/s000390050074**[E1]**Lawrence C. Evans,*Some new PDE methods for weak KAM theory*, Calc. Var. Partial Differential Equations**17**(2003), no. 2, 159–177. MR**1986317**, 10.1007/s00526-002-0164-y**[E2]**Lawrence C. Evans,*Towards a quantum analog of weak KAM theory*, Comm. Math. Phys.**244**(2004), no. 2, 311–334. MR**2031033**, 10.1007/s00220-003-0975-5**[F]**A. Fathi,*Weak KAM theorem in Lagrangian dynamics*, Cambridge University Press, in press.**[G1]**Diogo Aguiar Gomes,*A stochastic analogue of Aubry-Mather theory*, Nonlinearity**15**(2002), no. 3, 581–603. MR**1901094**, 10.1088/0951-7715/15/3/304**[G2]**Diogo Aguiar Gomes,*Generalized Mather problem and selection principles for viscosity solutions and Mather measures*, Adv. Calc. Var.**1**(2008), no. 3, 291–307. MR**2458239**, 10.1515/ACV.2008.012**[JKM]**H. R. Jauslin, H. O. Kreiss, and J. Moser,*On the forced Burgers equation with periodic boundary conditions*, Differential equations: La Pietra 1996 (Florence), Proc. Sympos. Pure Math., vol. 65, Amer. Math. Soc., Providence, RI, 1999, pp. 133–153. MR**1662751**, 10.1090/pspum/065/1662751**[LPV]**P. L. Lions, G. Papanicolaou, S. R. S. Varadhan,*Homogenization of Hamilton-Jacobi equations*, unpublished manuscript, circa 1988.**[M]**John N. Mather,*Action minimizing invariant measures for positive definite Lagrangian systems*, Math. Z.**207**(1991), no. 2, 169–207. MR**1109661**, 10.1007/BF02571383**[Y1]**Y. Yu,*variational problems, Aronsson equations and weak KAM theory*, Ph.D. dissertation, University of California, Berkeley, 2005.**[Y2]**Yifeng Yu,*A remark on the semi-classical measure from -\frac{ℎ²}2Δ+𝑉 with a degenerate potential 𝑉*, Proc. Amer. Math. Soc.**135**(2007), no. 5, 1449–1454 (electronic). MR**2276654**, 10.1090/S0002-9939-06-08702-8**[Y3]**Yifeng Yu,*𝐿^{∞} variational problems and weak KAM theory*, Comm. Pure Appl. Math.**60**(2007), no. 8, 1111–1147. MR**2330625**, 10.1002/cpa.20173

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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 Sánchez-Morgado**

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:
https://doi.org/10.1090/S0002-9939-10-10361-X

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/FCT-PT (PTDC/EEA-ACR/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.