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Pathological solutions to the Euler-Lagrange equation and existence/regularity of minimizers in one-dimensional variational problems. / Gratwick, Richard; Sedipkov, Aidys; Sychev, Mikhail и др.

в: Comptes Rendus Mathematique, Том 355, № 3, 01.03.2017, стр. 359-362.

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Gratwick R, Sedipkov A, Sychev M, Tersenov A. Pathological solutions to the Euler-Lagrange equation and existence/regularity of minimizers in one-dimensional variational problems. Comptes Rendus Mathematique. 2017 март 1;355(3):359-362. doi: 10.1016/j.crma.2017.01.020

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Gratwick, Richard ; Sedipkov, Aidys ; Sychev, Mikhail и др. / Pathological solutions to the Euler-Lagrange equation and existence/regularity of minimizers in one-dimensional variational problems. в: Comptes Rendus Mathematique. 2017 ; Том 355, № 3. стр. 359-362.

BibTeX

@article{338bebadeff140ee801fda7909f6f3a7,
title = "Pathological solutions to the Euler-Lagrange equation and existence/regularity of minimizers in one-dimensional variational problems",
abstract = "In this paper, we prove that if L(x,u,v)∈C3(R3→R), Lvv>0 and L≥α|v|+β, α>0, then all problems (1), (2) admit solutions in the class W1,1[a,b], which are in fact C3-regular provided there are no pathological solutions to the Euler equation (5). Here u∈C3[c,d[ is called a pathological solution to equation (5) if the equation holds in [c,d[, |u˙(x)|→∞ as x→d, and ‖u‖C[c,d]<∞. We also prove that the lack of pathological solutions to the Euler equation results in the lack of the Lavrentiev phenomenon, see Theorem 9; no growth assumptions from below are required in this result.",
author = "Richard Gratwick and Aidys Sedipkov and Mikhail Sychev and Aris Tersenov",
year = "2017",
month = mar,
day = "1",
doi = "10.1016/j.crma.2017.01.020",
language = "English",
volume = "355",
pages = "359--362",
journal = "Comptes Rendus Mathematique",
issn = "1631-073X",
publisher = "Elsevier",
number = "3",

}

RIS

TY - JOUR

T1 - Pathological solutions to the Euler-Lagrange equation and existence/regularity of minimizers in one-dimensional variational problems

AU - Gratwick, Richard

AU - Sedipkov, Aidys

AU - Sychev, Mikhail

AU - Tersenov, Aris

PY - 2017/3/1

Y1 - 2017/3/1

N2 - In this paper, we prove that if L(x,u,v)∈C3(R3→R), Lvv>0 and L≥α|v|+β, α>0, then all problems (1), (2) admit solutions in the class W1,1[a,b], which are in fact C3-regular provided there are no pathological solutions to the Euler equation (5). Here u∈C3[c,d[ is called a pathological solution to equation (5) if the equation holds in [c,d[, |u˙(x)|→∞ as x→d, and ‖u‖C[c,d]<∞. We also prove that the lack of pathological solutions to the Euler equation results in the lack of the Lavrentiev phenomenon, see Theorem 9; no growth assumptions from below are required in this result.

AB - In this paper, we prove that if L(x,u,v)∈C3(R3→R), Lvv>0 and L≥α|v|+β, α>0, then all problems (1), (2) admit solutions in the class W1,1[a,b], which are in fact C3-regular provided there are no pathological solutions to the Euler equation (5). Here u∈C3[c,d[ is called a pathological solution to equation (5) if the equation holds in [c,d[, |u˙(x)|→∞ as x→d, and ‖u‖C[c,d]<∞. We also prove that the lack of pathological solutions to the Euler equation results in the lack of the Lavrentiev phenomenon, see Theorem 9; no growth assumptions from below are required in this result.

UR - http://www.scopus.com/inward/record.url?scp=85012899981&partnerID=8YFLogxK

U2 - 10.1016/j.crma.2017.01.020

DO - 10.1016/j.crma.2017.01.020

M3 - Article

AN - SCOPUS:85012899981

VL - 355

SP - 359

EP - 362

JO - Comptes Rendus Mathematique

JF - Comptes Rendus Mathematique

SN - 1631-073X

IS - 3

ER -

ID: 10310287