【摘要】 本文研究满足一致椭圆型条件的散度型椭圆方程 ∑ni,j=1ddxjai,j(x) dudxi =0的很弱解 ,并以Hodge分解和弱逆H lder不等式为工具 ,证明了其正则性结果 :对任意的 2 - 2 n+ 1× 1 0 0 n2βα( 2 n+ 2 + 1 ) 2 -1<r≤ 2 ,都存在可积指数p =p n ,βα,r >2 ,使得对其任意很弱解u∈W1r,loc(Ω) ,都有u∈W1p ,loc(Ω) .特别 ,u是其通常意义下的弱解 .更多还原

【Abstract】 This paper considers very weak solutions of the divergence type elliptic equation ∑ni,j=1ddx ja i,j(x)dudx i=0,which satisfies the uniformly elliptic condition.The following regularity result is obtained by using Hodge decomposition and weakly reverse Hlder inequality:for any 2-2 n+1×100 n2βα(2 n+2+1)2 -1<r≤2,there exists integrable exponent p=pn,βα,r>such that any very weak solution u∈W 1,r loc(Ω) is in fact in W 1,p loc(Ω).In particular,u is a weak solution in the usual sense.更多还原