Estimates Of The Derivatives Of The Vector Fields

This paper devotes to the estimate of the Sobolev norm of the derivatives of the vector fields in a bounded domain in R³ in terms of their divergence and curl in the domain, and their normal or tangential components on the boundary. The estimates depend also on the topology of the domain. By using of the Hodge decomposition theorem of vector field, we establish the following results.Let G be a bounded domain in R³ with an unit outward normal v, (?)G∈C^k+4,1 < p <∞, k≥0 is an integer.Theorem 0.1 Suppose the second Betti number of G is m,m<∞. Then thereexists a constant C depending only on k,p,G, such that the following estimateholds:where g_i, i = 1,…,m, are linear functionals on L^p((?)G) .Theorem 0.2 Suppose the first Betti number of G is n, n <∞. Then thereexists a constant C depending only on k,p,G, such that the following estimateholds:where B_it, i = 1,…, n, are linear functionals on L^p((?)G).When the first Betti number and the second Betti number of the domain are finite, we have the following two inequalities: In this paper we also examine the minimization in the Banach space {u∈H¹(G)\{0}, v×u = 0} of the following functionalThe existence of the minimizers and the value of the infimum are discussed.