January  2005, 12(1): 97-114. doi: 10.3934/dcds.2005.12.97

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

1. 

CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal

2. 

Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal

Received  June 2003 Revised  August 2004 Published  December 2004

We look for necessary and sufficient conditions for the existence of solutions to the minimisation problem

$ (P) \qquad\qquad\qquad $ inf $\int_\Omega f $ (curl $u(x)) dx : u \in u_{\xi_0} + W^{1,\infty}_0(\Omega;\mathbb R^3)$

where the boundary data $u_{\xi_0}$ satisfies curl$u_{\xi_0}(x)= \xi_{0}$, for $\xi_0$ a given vector in $\mathbb R^3$.

Citation: Ana Cristina Barroso, José Matias. Necessary and sufficient conditions for existence of solutions of a variational problem involving the curl. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 97-114. doi: 10.3934/dcds.2005.12.97
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