一类非线性椭圆组很弱解的局部正则性

Local Regularity of Very Weak Solutions of a Class Nonlinear Elliptic Systems

讨论了 Rn( n≥ 2 )中有界开集Ω上二阶非线性椭圆组一 div A( x,u,Du) =B( x,u,Du) ,当 A( x,u,Du)满足强制与增长条件 ,B( x,u,Du)满足控制增长条件时 ,其很弱解 u( x)∈ W1 ,rloc(Ω ,Rn)的正则性 .其中 max{1 ,p - 1 }<r <p,p出现在A与 B的强制与增长假设中 .本文采用 Hodge分解的方法建立适当的检验函数 ,借助一些引理 ,对椭圆组的很弱解得到了逆 H lder不等式 ,从而改进了其很弱解偏微商的可积性 ,使其成为经典意义下的弱解 .

The regularity of very weak solutions u(x)∈W^(1,r) _(loc)(Ω,R^N) for a class nonlinear elliptic systems divA(x,u,Du)=B(x,u,Du) is discussed on bounded open set ΩR^n(n≥2), where A(x,u,Du) satisfies the coercive and growth conditions, B(x,u,Du) satisfies controllable growth condition, max{1,p-1}<r<p,p appears in the coercive and growth assumptions for the operator A and B. Here Hodge decomposition is used to construct a suitable test function, the reverse Holder inequality for very weak solutions of the elliptic systems is proved u(x) is improved, u(x) is a weak solution in usual sense.