The Mather problem for lower semicontinuous Lagrangians

Diogo A. Gomes; Gabriele Terrone

doi:10.1007/s00030-013-0243-0

The Mather problem for lower semicontinuous Lagrangians

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we develop the Aubry-Mather theory for Lagrangians in which the potential energy can be discontinuous. Namely we assume that the Lagrangian is lower semicontinuous in the state variable, piecewise smooth with a (smooth) discontinuity surface, as well as coercive and convex in the velocity. We establish existence of Mather measures, various approximation results, partial regularity of viscosity solutions away from the singularity, invariance by the Euler-Lagrange flow away from the singular set, and further jump conditions that correspond to conservation of energy and tangential momentum across the discontinuity. © 2013 Springer Basel.

Original language	English (US)
Pages (from-to)	167-217
Number of pages	51
Journal	Nonlinear Differential Equations and Applications NoDEA
Volume	21
Issue number	2
DOIs	https://doi.org/10.1007/s00030-013-0243-0
State	Published - Aug 1 2013

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: D. Gomes was partially supported by CAMGSD-LARSys through FCT-Portugal and by grants PTDC/MAT/114397/2009, UTAustin/MAT/0057/2008, and UTA-CMU/MAT/0007/2009.G.Terrone was supported by the UTAustin-Portugal partnership through the FCT post-doctoral fellowship SFRH/BPD/40338/2007, CAMGSD-LARSys through FCT Program POCTI - FEDER and by grants PTDC/MAT/114397/2009, UTAustin/MAT/0057/2008, and UTA-CMU/MAT/0007/2009.

ASJC Scopus subject areas

Analysis
Applied Mathematics

Access to Document

10.1007/s00030-013-0243-0

Cite this

@article{d094c83725d743b2a2d840f43f3eebaa,

title = "The Mather problem for lower semicontinuous Lagrangians",

abstract = "In this paper we develop the Aubry-Mather theory for Lagrangians in which the potential energy can be discontinuous. Namely we assume that the Lagrangian is lower semicontinuous in the state variable, piecewise smooth with a (smooth) discontinuity surface, as well as coercive and convex in the velocity. We establish existence of Mather measures, various approximation results, partial regularity of viscosity solutions away from the singularity, invariance by the Euler-Lagrange flow away from the singular set, and further jump conditions that correspond to conservation of energy and tangential momentum across the discontinuity. {\textcopyright} 2013 Springer Basel.",

author = "Gomes, {Diogo A.} and Gabriele Terrone",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: D. Gomes was partially supported by CAMGSD-LARSys through FCT-Portugal and by grants PTDC/MAT/114397/2009, UTAustin/MAT/0057/2008, and UTA-CMU/MAT/0007/2009.G.Terrone was supported by the UTAustin-Portugal partnership through the FCT post-doctoral fellowship SFRH/BPD/40338/2007, CAMGSD-LARSys through FCT Program POCTI - FEDER and by grants PTDC/MAT/114397/2009, UTAustin/MAT/0057/2008, and UTA-CMU/MAT/0007/2009.",

year = "2013",

month = aug,

day = "1",

doi = "10.1007/s00030-013-0243-0",

language = "English (US)",

volume = "21",

pages = "167--217",

journal = "Nonlinear Differential Equations and Applications NoDEA",

issn = "1021-9722",

publisher = "Birkhauser Verlag Basel",

number = "2",

}

TY - JOUR

T1 - The Mather problem for lower semicontinuous Lagrangians

AU - Gomes, Diogo A.

AU - Terrone, Gabriele

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: D. Gomes was partially supported by CAMGSD-LARSys through FCT-Portugal and by grants PTDC/MAT/114397/2009, UTAustin/MAT/0057/2008, and UTA-CMU/MAT/0007/2009.G.Terrone was supported by the UTAustin-Portugal partnership through the FCT post-doctoral fellowship SFRH/BPD/40338/2007, CAMGSD-LARSys through FCT Program POCTI - FEDER and by grants PTDC/MAT/114397/2009, UTAustin/MAT/0057/2008, and UTA-CMU/MAT/0007/2009.

PY - 2013/8/1

Y1 - 2013/8/1

N2 - In this paper we develop the Aubry-Mather theory for Lagrangians in which the potential energy can be discontinuous. Namely we assume that the Lagrangian is lower semicontinuous in the state variable, piecewise smooth with a (smooth) discontinuity surface, as well as coercive and convex in the velocity. We establish existence of Mather measures, various approximation results, partial regularity of viscosity solutions away from the singularity, invariance by the Euler-Lagrange flow away from the singular set, and further jump conditions that correspond to conservation of energy and tangential momentum across the discontinuity. © 2013 Springer Basel.

AB - In this paper we develop the Aubry-Mather theory for Lagrangians in which the potential energy can be discontinuous. Namely we assume that the Lagrangian is lower semicontinuous in the state variable, piecewise smooth with a (smooth) discontinuity surface, as well as coercive and convex in the velocity. We establish existence of Mather measures, various approximation results, partial regularity of viscosity solutions away from the singularity, invariance by the Euler-Lagrange flow away from the singular set, and further jump conditions that correspond to conservation of energy and tangential momentum across the discontinuity. © 2013 Springer Basel.

UR - http://hdl.handle.net/10754/562896

UR - http://link.springer.com/10.1007/s00030-013-0243-0

UR - http://www.scopus.com/inward/record.url?scp=84897064217&partnerID=8YFLogxK

U2 - 10.1007/s00030-013-0243-0

DO - 10.1007/s00030-013-0243-0

M3 - Article

SN - 1021-9722

VL - 21

SP - 167

EP - 217

JO - Nonlinear Differential Equations and Applications NoDEA

JF - Nonlinear Differential Equations and Applications NoDEA

IS - 2

ER -

The Mather problem for lower semicontinuous Lagrangians

Abstract

Bibliographical note

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this