NECESSARY AND SUFFICIENT CONDITIONS FOR

DISCRETE AND CONTINUOUS
DYNAMICAL SYSTEMS
Volume 12, Number 1, January 2005
Website: http://aimSciences.org
pp. 97–114
NECESSARY AND SUFFICIENT CONDITIONS FOR
EXISTENCE OF SOLUTIONS
OF A VARIATIONAL PROBLEM INVOLVING THE CURL
Ana Cristina Barroso
CMAF, Universidade de Lisboa
Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal
and
Departamento de Matemática
Faculdade de Ciências, Universidade de Lisboa
1749-016 Lisboa, Portugal
José Matias
Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1
1049-001 Lisboa, Portugal
(Communicated by N.S. Papageorgiou)
Abstract. We look for necessary and sufficient conditions for the existence of solutions to the minimisation problem
(P )
inf
Ω
f (curlu(x)) dx : u ∈ uξ0 + W01,∞ (Ω; R3 )
where the boundary data uξ0 satisfies curluξ0 (x) = ξ0 , for ξ0 a given vector in R3 .
1. Introduction. The search for minimisers of
1,∞
f (∇u(x)) dx : u ∈ u0 + W0 (Ω) ,
inf
Ω
when the integrand function f is non convex, has been undertaken extensively in the
literature (see, for example, [1], [2], [5], [6] and the references therein). Dacorogna
and Marcellini [5] showed that a necessary condition for the existence of solutions
to this problem is that the convex envelope of f , f ∗∗ , is globally affine.
In this paper we look at a similar question with the curl operator replacing the
gradient. Precisely, we identify necessary and sufficient conditions for the existence
of solutions to the problem
1,∞
3
f (curlu(x)) dx : u ∈ uξ0 + W0 (Ω; R )
(P )
inf
Ω
where the boundary data uξ0 has constant curl, that is, curluξ0 (x) = ξ0 , for ξ0 a
given vector in R3 .
Contrary to the gradient case, for the problem involving the curl we have one
more degree of freedom, which is expressed by the fact that our necessary condition
states that f ∗∗ is not strictly convex in at least two directions (cf. Definition 4.1),
as opposed to globally affine.
1991 Mathematics Subject Classification. 49J45, 49K24, 35E99.
Key words and phrases. differential inclusions, curl, convex envelope.
Research partially supported by FCT (Portugal).
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