Partial Differential Equations And End-point Estimates Involving Operators Div And Curl

This dissertation is devoted to studying existence,regularity and Liouville type theorem of several PDE models in mathematical physics involving opera-tors div and curl and establishing Hardy type inequalities for vector fields and boundedness of a class of fractional integral operators in some proper spaces.In the first chapter,we briefly introduce our research motivations and state the main results.In the second chapter,we study a time-independent thermoelectric model,which is a coupled system of the Maxwell equations and an elliptic equation.Existence and regularity of weak solutions are obtained for general boundary data,while uniqueness is established for small boundary data.Several related systems are also studied.In the third chapter,we study the Beltrami flows in both bounded and unbounded domains by establishing an elementary but useful identity involving operators div and curl.In the case of bounded domains,we show that a Bel-trami flow in a star-shaped domain with vanishing tangential component is in fact trivial.In the case of unbounded domains,under the decay assumption|u(x)| = o(|x|-1)as |x|?+?,we prove that a Beltrami flow in a star-shaped unbounded domain with vanishing tangential component on the boundary or in the complement of the closure of a star-shaped domain with vanishing nor-mal component on the boundary is necessarily trivial.The same method is also applied to study the Maxwell and Stokes eigenvalue problems.We show that for a bounded and star-shaped domain the first Maxwell eigenvalue under nor-mal boundary condition or tangent boundary condition is strictly smaller than the first Stokes eigenvalue.Existence of solutions to the Hall-MHD system in three-dimensional bounded domains is also proved.Under the condition of small external forces and using the Schauder's fixed point theorem,we get existence of a weak solution to the Hall-MHD system with Holder continuous magnetic field.In the final chapter,we first study the fractional integral operator for vector fields in the Ll and weighted L1 spaces.Then we give a new and elementary approach to an inequality of Bourgain and Brezis for L1 vector fields.Finally,we establish Hardy-type inequalities for vector fields involving the L1 norm of the operator curl.We also prove the Hardy-type inequalities on bounded domains and for non-divergence-free vector fields with tangential components vanishing on the boundary.