Minimum Of Variation Problem With Operator Curl

0.2 ABSTRACTThis paper devotes to the minimization problem of a functional involving the operator curl of vector fieldsWe discuss existence and regularity of the minimizers of the functional J[u] with the vector field subjected to one of the following boundary conditions: i) uT=g, ii)u·v=g,whereΩis a bounded domain in R3, uT denotes the tangential component of u on the boundary (?)Ω,and v is the unit outer normal of (?)Ω.So the conditionⅰ)is prescribing the tangential component of the vector fields,andⅱ)is prescribing the normal component of the vector fields.We shall mainly consider the first boundary condition,as the methods used are applicable as well to the problem under the second boundary condition.Under some conditions on the function f we obtain the existence of minimizers,and give a description of all minimizers when p≥6/5.In the case where p=2 we establish the H2 regularity and the estimate for the minimizers. We shall also consider the variational problem with a point-wise constraint...