The Properties Of The Solutions For Elliptic Equation And Its Obstacle Problem

In the research of quasilinear elliptic partial differential equations of second order，it is veryimportant to investigate the well-posedness of solutions of equations. In fluid mechanics andengineering there is a wide range of applications about the well-posedness of solutions. Above all,A-harmonic equation is widely applied in quasiregular mappings, elasticity and physics. Hence,there is very importance significance to research the well-posedness of solutions for this kind ofequations.Well-posedness of solutions include existence, uniqueness and stability. At present, the weakand very weak solutions of quasilinear elliptic equations of second order have been studiedextensively in different spaces and different conditions. The stability of solutions is to say thecontinuous dependence of solutions, which is based on the existence and uniqueness of solutions.In view of the known results, we study the stability of weak solutions for a class of quasilinearelliptic equations with varying p-index, as well as the local and global regularity of very weaksolutions of obstacle problems to a class of elliptic equations. About stability of solutions regardingp-index, we get the results based on the uniformly estimate to the gradient of weak solutions byusing Sobolev inequalities with uniformly p-thick boundary, H lder’s and Young’s inequality.Under the conditions of weaker controllable growth with respect to u and Du in B ( x, u , Du ), weapply the H lder’s and Young’s inequality twice for the same integrand. Similar to the methodmentioned above in the local regularity and considering that the boundaryΩis r-Poincaréthick,we obtain the global regularity of very weak solutions of obstacle problems by Poincaré’s inequality,and Minkowski’s inequality.This paper consists of four parts. In the first chapter, we briefly relate the background andhistory of solutions of elliptic equations and relevant obstacle problems. In the second chapter weconsider stability of weak solutions with varying p-index for a class of quasilinear elliptic equations.In the third chapter, we obtain local regularity for very weak solutions of obstacle problems to aclass of elliptic equations. In the last chapter we get global regularity for very weak solutions ofobstacle problems. We mainly use the method of Sobolev space analysis, nonlinear potential theory,apply Hodge decomposition, reverse H lder’s inequality and others in our work.