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

Richard Gratwick, Aidys Sedipkov, Mikhail Sychev, Aris Tersenov

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

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.

Original languageEnglish
Pages (from-to)359-362
Number of pages4
JournalComptes Rendus Mathematique
Volume355
Issue number3
DOIs
Publication statusPublished - 1 Mar 2017

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