论文标题
椭圆形和抛物线运营商的结构定理，并应用于Kolmogorov类型的操作员的同质化
A structure theorem for elliptic and parabolic operators with applications to homogenization of operators of Kolmogorov type
论文作者
论文摘要
我们认为操作员\ [ \ nabla_x \ cdot（a（x）\ nabla_x），\ \ \ nabla_x \ cdot（a（x）\ nabla_x） - \ partial_t，\ \ \ \ \ \ \ nabla_x \ cdot（a（x）\ nabla_x） ω$，$（x，t）\ inω\ times \ times \ mathbb r $和$（x，y，t）\ inω\ times \ times \ mathbb r^m \ times \ times \ times \ times \ mathbb r $，其中$ω\ subset \ subset \ subset \ subset \ subbb r^m $是（无界）lipchitz domain comphitz domain defiar $ nath $ nath r^{m-1} \ to \ mathbb r $是Lipschitz，由$ m $限制。假设相对于表面度量$ \ mathrm {d}σ（x）$，与第一个操作员相关的椭圆度度量是绝对连续的，并且相应的radon-nikodym derivativative或poisson内核满足了规模不变的反向höld在$ l^p $中，仅在$ l^p $中，$ 1 $ p $ p <。 $ a $，$ m $和$ψ$，$ m $的Lipschitz常数。在这个假设下，我们证明，与第二和第三操作员相关的抛物面度量也是如此，$ \ mathrm {d}σ（x）$被表面度量代替了$ \ mathrm {d}σ（d}σ（x）\ mathrm {d} t $ {d} t $和$ \ mathrm {d} d} d}} y \ mathrm {d} t $。这种结构定理使我们能够确定先前在文献中建立的几个结果，并在例如Kolmogorov类型运营商的同质化背景下推断出新的结果。我们对结构定理的证明是基于作者最新的结果，该结果涉及Kolmogorov类型的运营商的边界harnack不平等，并具有有限的，可测量的和均匀的椭圆系数。
We consider the operators \[ \nabla_X\cdot(A(X)\nabla_X),\ \nabla_X\cdot(A(X)\nabla_X)-\partial_t,\ \nabla_X\cdot(A(X)\nabla_X)+X\cdot\nabla_Y-\partial_t, \] where $X\in Ω$, $(X,t)\in Ω\times \mathbb R$ and $(X,Y,t)\in Ω\times \mathbb R^m\times \mathbb R$, respectively, and where $Ω\subset\mathbb R^m$ is a (unbounded) Lipschitz domain with defining function $ψ:\mathbb R^{m-1}\to\mathbb R$ being Lipschitz with constant bounded by $M$. Assume that the elliptic measure associated to the first of these operators is mutually absolutely continuous with respect to the surface measure $\mathrm{d} σ(X)$, and that the corresponding Radon-Nikodym derivative or Poisson kernel satisfies a scale invariant reverse Hölder inequality in $L^p$, for some fixed $p$, $1<p<\infty$, with constants depending only on the constants of $A$, $m$ and the Lipschitz constant of $ψ$, $M$. Under this assumption we prove that then the same conclusions are also true for the parabolic measures associated to the second and third operator with $\mathrm{d} σ(X)$ replaced by the surface measures $\mathrm{d} σ(X)\mathrm{d} t$ and $\mathrm{d} σ(X)\mathrm{d} Y\mathrm{d} t$, respectively. This structural theorem allows us to reprove several results previously established in the literature as well as to deduce new results in, for example, the context of homogenization for operators of Kolmogorov type. Our proof of the structural theorem is based on recent results established by the authors concerning boundary Harnack inequalities for operators of Kolmogorov type in divergence form with bounded, measurable and uniformly elliptic coefficients.
