论文标题
带有BMO反对称部分的二阶椭圆算子的摄动理论
Perturbation Theory for Second Order Elliptic Operators with BMO Antisymmetric Part
论文作者
论文摘要
在本文中，我们研究了针对椭圆形算子的$ l^p $ dirichlet问题的扰动理论，以发散形式为椭圆形算子，在BMO中具有潜在无限的反对称部分。具体而言，给定椭圆运算符$ l_0 = \ mbox {div}（a_0 \ nabla）$和$ l_1 = \ mbox {div}（a_1 \ nabla）$，使得$ l^p $ dirichlet问题用于$ l_0 $ l_0 $可解决一些$ p> 1 $;我们表明，如果$ a_0 -a_1 $满足某些Carleson条件，则$ l^q $ dirichlet问题$ l_1 $对于某些$ q \ geq p $可解决。此外，如果Carleson Norm很小，那么我们可能会服用$ Q = P $。我们使用该方法在单位球上首次引入的Fefferman-Kenig-Pipher '91，并建立在Milakis-Pipher-Toro '11上，其中显示了有界和弦弧形域上的对称矩阵的大型尺寸。 We then apply this to solve the $L^p$ Dirichlet problem on a bounded Lipschitz domain for an operator $L = \mbox{div}(A\nabla)$, where $A$ satisfies a Carleson condition similar to the one assumed in Kenig-Pipher '01 and Dindoš-Petermichl-Pipher '07 but with unbounded antisymmetric part.
In the present paper we study perturbation theory for the $L^p$ Dirichlet problem on bounded chord arc domains for elliptic operators in divergence form with potentially unbounded antisymmetric part in BMO. Specifically, given elliptic operators $L_0 = \mbox{div}(A_0\nabla)$ and $L_1 = \mbox{div}(A_1\nabla)$ such that the $L^p$ Dirichlet problem for $L_0$ is solvable for some $p>1$; we show that if $A_0 - A_1$ satisfies certain Carleson condition, then the $ L^q$ Dirichlet problem for $L_1$ is solvable for some $q \geq p$. Moreover if the Carleson norm is small then we may take $q=p$. We use the approach first introduced in Fefferman-Kenig-Pipher '91 on the unit ball, and build on Milakis-Pipher-Toro '11 where the large norm case was shown for symmetric matrices on bounded chord arc domains. We then apply this to solve the $L^p$ Dirichlet problem on a bounded Lipschitz domain for an operator $L = \mbox{div}(A\nabla)$, where $A$ satisfies a Carleson condition similar to the one assumed in Kenig-Pipher '01 and Dindoš-Petermichl-Pipher '07 but with unbounded antisymmetric part.
