论文标题
针对可满足Carleson条件的真实系数的运营商的规律性和Neumann问题
Regularity and Neumann problems for operators with real coefficients satisfying Carleson condition
论文作者
论文摘要
In this paper, we continue the study of a class of second order elliptic operators of the form $\mathcal L=\mbox{div}(A\nabla\cdot)$ in a domain above a Lipschitz graph in $\mathbb R^n,$ where the coefficients of the matrix $A$ satisfy a Carleson measure condition, expressed as a condition on the oscillation on Whitney balls.对于这类运营商来说，众所周知（自2001年以来），$ l^q $ dirichlet问题可解决一些$ 1 <q <\ infty $。此外，当Carleson测量振动范围足够小时，进一步的研究完全解决了Dirichlet，Lipschitz域中的$ L^Q $溶解性的范围。 我们表明存在$ p_ {reg}> 1 $，以便所有$ 1 <p <p <p <p_ {reg} $ $ l^p $常规性问题对于操作员$ \ mathcal $ \ mathcal l = \ mbox {div}（a \ nabla \ cdot）$可解决。此外，$ \ frac1 {p_ {reg}}+\ frac1 {q _*} = 1 $其中$ q _*> 1 $是数字，以至于$ \ mathcal $ \ mathcal l^*$可解决所有$ q> q q _*$。 此外，当$ n = 2 $时，存在$ p_ {neum}> 1 $，以便对于所有$ 1 <p <p <p <p <p <p <p_ {neum} $ $ l^p $ neumann问题，用于操作员$ \ mathcal $ \ mathcal l = \ mbox {div}（a \ nabla \ cdot）$溶解。此外可解决所有$ q> q^*$。
In this paper, we continue the study of a class of second order elliptic operators of the form $\mathcal L=\mbox{div}(A\nabla\cdot)$ in a domain above a Lipschitz graph in $\mathbb R^n,$ where the coefficients of the matrix $A$ satisfy a Carleson measure condition, expressed as a condition on the oscillation on Whitney balls. For this class of operators, it is known (since 2001) that the $L^q$ Dirichlet problem is solvable for some $1 < q < \infty$. Moreover, further studies completely resolved the range of $L^q$ solvability of the Dirichlet, Regularity, Neumann problems in Lipschitz domains, when the Carleson measure norm of the oscillation is sufficiently small. We show that there exists $p_{reg}>1$ such that for all $1<p<p_{reg}$ the $L^p$ Regularity problem for the operator $\mathcal L=\mbox{div}(A\nabla\cdot)$ is solvable. Furthermore $\frac1{p_{reg}}+\frac1{q_*}=1$ where $q_*>1$ is the number such that the $L^q$ Dirichlet problem for the adjoint operator $\mathcal L^*$ is solvable for all $q>q_*$. Additionally when $n=2$, there exists $p_{neum}>1$ such that for all $1<p<p_{neum}$ the $L^p$ Neumann problem for the operator $\mathcal L=\mbox{div}(A\nabla\cdot)$ is solvable. Furthermore $\frac1{p_{reg}}+\frac1{q^*}=1$ where $q^*>1$ is the number such that the $L^q$ Dirichlet problem for the operator $\mathcal L_1=\mbox{div}(A_1\nabla\cdot)$ with matrix $A_1=A/\det{A}$ is solvable for all $q>q^*$.
