【作者】 梁爽；

【导师】 高红亚；

【作者基本信息】 河北大学 ， 数学， 2016， 硕士

【摘要】 本文分为五章。第一章和第五章分别为引言和总结。第二章讨论各项异性椭圆方程解的全局可积性。这章中给出了边界值问题的弱解的全局可积性的新的证明,这个证明方法与经典方法是不同的。证明了边界值u*的更高的可积性保证了u也有更高的可积性。对积分泛函和障碍问题也得到类似结果。第三章考虑p-调和方程边值问题利用Hodge分解,给出在假设θ∈W1,q(Q),q>r下很弱解u的可积性。第四章考虑两种椭圆方程组的边值问题。4.1节考虑了非其次椭圆方程组的弱解u∈W1,(pi)(Q,RN),给出了矩阵a=(aiα)∈RN×n的单调不等式,保证了各项异性的解的全局逐点有界性。4.2节研究了非齐次拟线性椭圆方程组弱解u的正则性。假设在分量yγ很大时,非对角系数aijγβ(x,y)和∫iγ(x,y)足够小,得到对任意弱解u∈W1,2(Q,RN)都有u∈Lweak2*(1+q)(Ω,RN)。更多还原

【Abstract】 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 θ E W1,q(Ω), 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:Ω(?)Rn→RN of nonhomogeneous quasilinear elliptic systems. The diagonal coefficients aijγγ(x,y) and fiγ(x,y) are assumed to be small when the corresponding component yγ is large. We derive u∈Lweak2*(1+q) (Ω, RN) for every weak solution u∈W1,2(Ω,RN).更多还原