L~p Estimates Of The Vector Fields And Study On The Stability Of Meissner State For Superconductors

In this paper we consider the L3/2estimates of vector fields. We prove the L3/2norm of vector fields with the tangential components varnishing on the boundary can be controlled by the W-1//3.3/2norm of divergence, the L1norm of the curl and some terms depending on the topology of the domain. By a similar discussion, we also establish the general Lp estimates for1<p<∞. As the application of the Lp estimates, we prove the Global div-curl lemma in Sobolev spaces of negative indices.As the application of the estimates of the vector fields, we also study a partial differential system involving the operator curl which was derived from the mathe-matical theory of superconductivity. We obtain the following results:for the radially symmetric domain in R2, we prove the existence of the unstable solution and get the estimate of the critical field, some properties of the symmetric solutions are also studied; by the variational method, we show the existence and regularity of the solutions in the exterior domains in R3, and derive the exponential decay of the so-lution at infinity; for the bounded3-dimensional superconductor, when the applied magnetic field increases, we consider the location of the maximal points of the mag-netic potential of induced magnetic field, for the small penetration depth and under a uniform estimate assumption, we prove that the location of the maximal points of the magnetic potential of induced magnetic field depends on the points that the maximum of the tangential component of the applied magnetic field attained and the normal curvature of the boundary.