Necessary and sufficient conditions for existence of solutions of a variational problem involving the curl

Abstract

We look for necessary and sufficient conditions for the existence of solutions to the minimisation problemwhere the boundary data u ξ 0 satisfies curlu ξ 0 (x) = ξ 0 , for ξ 0 a given vector in R 3 .

Differential inclusions for differential forms

“…When dim span E = 2, Theorem 4.15 is just a restatement of Theorem 4.13 because we can take F = E. However, this sufficient condition is also true when dim span E = 3 so this theorem improves the aforementioned one. It also extends previous results of [17], which do not consider Dirichlet boundary conditions, and of [3], [9], where a non-constructive proof was obtained only for the case dim span F = 2.…”

“…This result improves Barroso-Matias [3], Dacorogna-Fonseca [9] and JamesKinderlehrer [17], see Remark 4.16 for details.…”

“…This is a particularly important problem for applications, such as the Ginzburg-Landau model of ferromagnetism, and we refer to Barroso-Matias [3], Dacorogna-Fonseca [9], DeSimoneDolzmann [11], James-Kinderlehrer [17] and to the references therein.…”

Minimizing curl in a multiconnected domain

“…ᮀ One may consider more general minimization problems for curl of vector fields, for example, to minimize the L p norm of curl or to minimize a more general functional of curl. 1 …”

“…1 In this paper we consider Problem 1.1 in the Sobolev space H 1 ͑⍀ , R 3 ͒. To state this problem more precisely, let ⍀ be a bounded domain in R 3 with boundary ‫ץ‬⍀ being of class C r , r Ն 2, and let H be a given vector field defined on ‫ץ‬⍀.…”