Null Lagrangians, weak continuity, and variational problems of arbitrary order

J. M. Ball; J. C. Currie; P. J. Olver

doi:10.1016/0022-1236(81)90085-9

Null Lagrangians, weak continuity, and variational problems of arbitrary order

J. M. Ball, J. C. Currie, P. J. Olver

Mathematics

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Abstract

We consider the problem of minimizing integral functionals of the form I(u) = ∝_Ω F(x, ▽^[k]u(x)) dx, where Ω ⊂R^p, u:ω →R and ▽^[k]u denotes the set of all partial derivatives of u with orders ≤k. The method is based on a characterization of null Lagrangians L(▽^ku) depending only on derivatives of order k. Applications to elasticity and other theories of mechanics are given.

Original language	English (US)
Pages (from-to)	135-174
Number of pages	40
Journal	Journal of Functional Analysis
Volume	41
Issue number	2
DOIs	https://doi.org/10.1016/0022-1236(81)90085-9
State	Published - Apr 1981

Bibliographical note

Funding Information:
The research of J.M.B. was partially supported by U.S. Army Contract DAAG20-79-C 0086, that of J.C.C. by a United Kingdom S.R.C. research studentship, and that of P.J.O. by a United Kingdom S.R.C. research grant.

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title = "Null Lagrangians, weak continuity, and variational problems of arbitrary order",

abstract = "We consider the problem of minimizing integral functionals of the form I(u) = ∝Ω F(x, ▽[k]u(x)) dx, where Ω ⊂Rp, u:ω →R and ▽[k]u denotes the set of all partial derivatives of u with orders ≤k. The method is based on a characterization of null Lagrangians L(▽ku) depending only on derivatives of order k. Applications to elasticity and other theories of mechanics are given.",

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note = "Funding Information: The research of J.M.B. was partially supported by U.S. Army Contract DAAG20-79-C 0086, that of J.C.C. by a United Kingdom S.R.C. research studentship, and that of P.J.O. by a United Kingdom S.R.C. research grant.",

year = "1981",

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AU - Ball, J. M.

AU - Currie, J. C.

AU - Olver, P. J.

N1 - Funding Information: The research of J.M.B. was partially supported by U.S. Army Contract DAAG20-79-C 0086, that of J.C.C. by a United Kingdom S.R.C. research studentship, and that of P.J.O. by a United Kingdom S.R.C. research grant.

PY - 1981/4

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N2 - We consider the problem of minimizing integral functionals of the form I(u) = ∝Ω F(x, ▽[k]u(x)) dx, where Ω ⊂Rp, u:ω →R and ▽[k]u denotes the set of all partial derivatives of u with orders ≤k. The method is based on a characterization of null Lagrangians L(▽ku) depending only on derivatives of order k. Applications to elasticity and other theories of mechanics are given.

AB - We consider the problem of minimizing integral functionals of the form I(u) = ∝Ω F(x, ▽[k]u(x)) dx, where Ω ⊂Rp, u:ω →R and ▽[k]u denotes the set of all partial derivatives of u with orders ≤k. The method is based on a characterization of null Lagrangians L(▽ku) depending only on derivatives of order k. Applications to elasticity and other theories of mechanics are given.

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Null Lagrangians, weak continuity, and variational problems of arbitrary order

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