论文标题
具有粗糙系数的Kolmogorov类型的一类强烈退化的抛物线算子的潜在理论
Potential theory for a class of strongly degenerate parabolic operators of Kolmogorov type with rough coefficients
论文作者
论文摘要
在本文中，我们开发了一种潜在理论，用于形式的强烈退化抛物面操作员：= \ nabla_x \ cdot（a（x，x，x，x，y，t）\ nabla_x）+x \ cdot \ cdot \ cdot \ cdot \ nabla_ {y} {y} - potial_t，\ partial_t，\ paintial_t，\ \ \ \ \ \ \ \ \ \ \ \ \] ω= \ {（x，x，y，t）=（x，x，x_ {m}，y，y，y_ {m}，t）\ in \ mathbb r^{m-1} \ times \ times \ times \ times \ times \ times \ times \ times \ times \ times \ mathbb r^{m-1} x_m>ψ（x，y，y_m，t）\}，\]，其中假定$ψ$满足适合扩张结构的均匀Lipschitz条件，而（非欧巴文）则是操作员$ \ \ \ mathcal {l} $的（non-euclidean）。关于$ a = a = a（x，y，t）$，我们假设$ a $是有界，可测量，对称和均匀椭圆形的（作为$ \ mathbb r^{m} $中的矩阵）。除了Dirichlet问题和其他基本属性的解决性之外，我们的结果还包括规模和翻译不变边界比较原理，边界Harnack的不等式以及相关抛物线措施的加倍属性。我们所有的估计都是翻译和比例不变的，常数仅取决于定义$ a $的界限和椭圆度和$ψ$的Lipschitz常数的常数。我们的结果代表了一个版本，适用于Kolmogorov类型的运营商，具有有界的，可测量的系数，现在是Fabes和Safonov的经典结果，即其他几个版本，涉及（时间依赖性）Lipschitz类型域中均匀抛物线方程的边界估计。
In this paper we develop a potential theory for strongly degenerate parabolic operators of the form \[ \mathcal{L}:=\nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_{Y}-\partial_t, \] in unbounded domains of the form \[ Ω=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{m-1}\times\mathbb R\times\mathbb R^{m-1}\times\mathbb R\times\mathbb R\mid x_m>ψ(x,y,y_m,t)\}, \] where $ψ$ is assumed to satisfy a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator $\mathcal{L}$. Concerning $A=A(X,Y,t)$ we assume that $A$ is bounded, measurable, symmetric and uniformly elliptic (as a matrix in $\mathbb R^{m}$). Beyond the solvability of the Dirichlet problem and other fundamental properties our results include scale and translation invariant boundary comparison principles, boundary Harnack inequalities and doubling properties of associated parabolic measures. All of our estimates are translation- and scale-invariant with constants only depending on the constants defining the boundedness and ellipticity of $A$ and the Lipschitz constant of $ψ$. Our results represent a version, for operators of Kolmogorov type with bounded, measurable coefficients, of the by now classical results of Fabes and Safonov, any several others, concerning boundary estimates for uniformly parabolic equations in (time-dependent) Lipschitz type domains.
