奇异对流方程组非常弱解的梯度正则性

Gradient Regularity of Very Weak Solution to Elliptic Equations with Singular Convection

该文主要考虑奇异对流方程组非常弱解的梯度部分正则性.首先,结合Lorentz空间及其与Lebesgue空间之间的关系,推出奇异对流方程组在L^(p)空间存在非常弱解.接着,通过Hodge分解证明Dirichlet问题的非常弱解实际上就是古典弱解.最后,利用A-调和逼近技巧,建立了奇异对流方程组非常弱解的梯度部分正则性结果,最重要的是,由此所得到的正则性结果是最优的.

This paper deals with the partial regularity of very weak solutions to elliptic equations with singular convective.By the properties of Lorentz space and its relation to Lebesgue space,we conclude that the elliptic systems with singular convection have very weak solutions in L^(p)space.Then,it can be found from Hodge decomposition that the very weak solutions of Dirichlet problem are actually the classical weak solutions.Finally,combining with A-harmonic approximation technique,we further find that the obtained weak solution has partial regularity;especially,the regularity is optimal.