Integrability For Solutions To Elliptic Equations And Systems

There are five chapters in this paper. Chapter 1 and 5 are introduction and conclusions, respectively. The second chapter considers boundary value problems of the form We show, by a different method from the classical ones, that higher integrability of the bound-ary datum-u* forces u to have higher integrability as well. Similar results are also obtained for obstacle problems and integral functional.In chapter 3, we deal with boundary value problems of p-harmonic equation We show, by Hodge decomposition,that under the assumption ? E W1,q(?), q>r, any very weak solution u to the boundary value problem is integrable.In chapter 4, we consider elliptic systems. In section 4.1, we deal with anisotropic solu-tions u?W1,(pi)(?, RN) to the nonlinear elliptic system We present a monotonicity inequality for the matrix a=(ai?) E RN×n, which guarantees global pointwise bounds for anisotropic solutions u. In section 4.2, we consider regularity properties for weak solutions u:?(?)Rn?RN of nonhomogeneous quasilinear elliptic systems. The diagonal coefficients aij??(x,y) and fi?(x,y) are assumed to be small when the corresponding component y? is large. We derive u?Lweak2*(1+q) (?, RN) for every weak solution u?W1,2(?,RN).