Interior Regularity To Degenerate Elliptic (Parabolic) Systems And Ultraparabolic Equations

Elliptic systems and parabolic systems have an important theoretical and practical significance in science and engineering fields. With the development of theory of partial differential equations, many systems constituted by Hormander’s vector fields received an extensive attention. Regularity for weak solutions to diagonal and nondiagonal elliptic and parabolic systems in Euclidean spaces has been treated by many authors. But regularity for weak solutions to nondiagonal degenerate elliptic and parabolic systems have not yet to see. This thesis considers (f) regularity for weak solutions to quasilinear nondiagonal degenerate elliptic and parabolic systems of smooth Hormander’s vector fields with the lower order terms satisfying some growth conditions; (2) regularity for weak solutions to a class of ultraparabolic equations. It involves the following three parts.The first part is concerned with regularity for weak solutions to the nondi-agonal quasilinear degenerate elliptic system of smooth Hormander’s vector fields. To do so, the system is divided into two subsystems by the decomposition of co-efficients, i.e., a diagonal homogeneous system and a diagonal nonhomogeneous system; then Lp(p≥2) estimates for gradients of weak solutions to these two sub-systems are obtained, respectively. With them and the reverse Holder inequality on the homogeneous space, estimates of gradients of weak solutions to nondiagonal degenerate elliptic system are got. By considering regularity for weak solutions to homogeneous and nonhomogeneous systems, integrability for gradient of weak so-lutions to the nonhomogeneous degenerate elliptic system is proved, higher Morrey (Lp’λ) regularity for gradients of weak solutions to nondiagonal degenerate elliptic system with some growth conditions is established, and then Campanato regu-larity for weak solutions is got. Holder regularity with exact Holder index for weak solutions is deduced by Morrey lemma. When low terms satisfy another growth conditions, higher Campanato regularity for gradients of weak solutions is investigated.Regularity for weak solutions to nondiagonal quasilinear parabolic systems of smooth Hormander’s vector fields is studied in the second part, where the low terms satisfy the natural growth conditions. Due to the lack of parabolic Poincare inequality and parabolic Sobolev inequality, the first task is establish these in-equalities. The parabolic Poincare inequality is proved by choosing appropriate cutoff functions. By introducing the average of weak solutions on measurable balls, the parabolic Sobolev inequality and Caccioppoli inequality are inspected. Combining these inequalities with the reverse Holder inequality on the homoge-neous space, higher Lp regularity and Morrey regularity for gradients of weak solutions are demonstrated. And then by the parabolic Poincare inequality, Cam-panato regularity for weak solutions is derived. Due to lack of parabolic Morrey lemma, Holder regularity for weak solutions is gained by proving the isomorphic relationship between the Campanato space and Holder space.The third part focus on regularity for weak solutions to a nonhomogeneous ultraparabolic equation. Related equations was studied by the singular integral method and a prior estimates method, respectively. Here a prior estimates method is used. When weak solutions belong to L2, integrability for weak solutions is lifted by the reverse Holder inequality on the homogeneous space, and then higher regu-larity including Lp estimates and Morrey estimates for gradients of weak solutions are obtained. In establishing higher regularity for gradients, the Sobolev inequality and the Poincare inequality are needed. The Sobolev inequality is proved by using properties of the fundamental solution to freezing operator of ultraparabolic opera-tor, and the Poincare inequality by choosing suitable cutoff function. Campanato regularity for weak solutions is followed by Morrey estimates and the Poincare inequality.