TY - JOUR

T1 - Variational field theory from the point of view of direct methods

AU - Sychev, M. A.

PY - 2017/9/1

Y1 - 2017/9/1

N2 - In this paper we show that the classical field theory ofWeierstrass–Hilbert can be strengthen on applying direct methods. Concretely, given a field of extremals and an extremal that is an element of the field, we can show that the latter gives minimum in the class of Lipschitz functions with the same boundary data and with the graphs in the set covered by the field. We suggest the two proofs: a modern one (exploiting Tonelli’s Theorem on lower semicontinuity of integral functionals with respect to the weak convergence of admissible functions in W1,1) and the one based only on arguments available already in the 19th century.

AB - In this paper we show that the classical field theory ofWeierstrass–Hilbert can be strengthen on applying direct methods. Concretely, given a field of extremals and an extremal that is an element of the field, we can show that the latter gives minimum in the class of Lipschitz functions with the same boundary data and with the graphs in the set covered by the field. We suggest the two proofs: a modern one (exploiting Tonelli’s Theorem on lower semicontinuity of integral functionals with respect to the weak convergence of admissible functions in W1,1) and the one based only on arguments available already in the 19th century.

KW - direct methods

KW - ellipticity

KW - Euler equation

KW - filed theory

KW - integral functionals

KW - minimizer

UR - http://www.scopus.com/inward/record.url?scp=85032015600&partnerID=8YFLogxK

U2 - 10.1134/S0037446617050160

DO - 10.1134/S0037446617050160

M3 - Article

AN - SCOPUS:85032015600

VL - 58

SP - 891

EP - 898

JO - Siberian Mathematical Journal

JF - Siberian Mathematical Journal

SN - 0037-4466

IS - 5

ER -