摘要
讨论了 Rn(n≥ 3)中有界区域 Ω上二阶非齐次拟线性椭圆型方程 - div A(x,u) =B(x,u) .当 A(x,u)满足控制增长条件和单调不等式 ,B(x,u)满足控制增长条件 |B(x,u) |≤C′| u|p-1时 ,其很弱解 u(x)∈ W1,rloc(Ω)的正则性 ,其中 max{1 ,p- 1 }<r<p,p为自然的 Sobolev空间指数 .文中采用 Hodge分解的方法建立试验函数 ,借助 Ho ¨lder不等式、Poincaré不等式及Young不等式对方程的很弱解得到了逆 Ho ¨lder不等式 ,从而改进了其很弱解偏微商的可积性 。
The regularity of very weak solutions u(x)∈W 1,r loc (Ω) for a nonhomogeneous quasilinear elliptic type second order equation -div A(x,FDA1u)=B(x,FDA1u) was discussed on a bounded domain ΩR n(n≥3) , where A(x, FDA1 u) satisfies the controllable growth condition and the monotonicity inequality, B(x, FDA1 u) satisfies the controllable growth condition ｜B(x, FDA1 u)｜≤C′｜ FDA1 u｜ p-1 , and max ｛1,p-1｝<r<p, p is a usual Sobolev space exponent. Here Hodge decomposition is used to construct a suitable test function, the reverse Ho ¨lder's inequality for very weak solutions of the equation is proved with the aid of Ho ¨lder's inequality, Poincaré's inequality and Young's inequality etc. Hence the integrability of the derivative of very weak solution u(x) is improved, u(x) is a weak solution in the usual sense.
出处
《上海交通大学学报》
EI
CAS
CSCD
北大核心
2001年第7期1105-1108,共4页
Journal of Shanghai Jiaotong University
基金
国家自然科学基金 (195 310 6 0 )
国家教委博士点基金资助项目 (970 2 4811)
