arXiv 2003.090372020-03math.APmath.FA

具有混合维数边界的区域中的椭圆理论

Elliptic theory in domains with boundaries of mixed dimension

Guy David, Joseph Feneuil, Svitlana Mayboroda

备注：: 116 pages. In version 2, we completed our theory with Green functions and a comparison principle

AI总结：

针对边界由不同维数片段组成的开区域，本文构造适应几何的退化椭圆算子，建立边界Poincaré、Harnack、最大值原理及Hölder连续性等估计，构造椭圆测度与Green函数并证明比较原理，推广Caffarelli-Sylvestre延拓算子及分数阶Laplacian。

摘要：

考虑一个开区域 $Ω\subset \mathbb R^n$，其边界可能由不同维数的部分组成。例如，$Ω$ 可以是 $\mathbb R^3$ 中的一个球去掉其一条直径 $D$，或者 $Ω\subset \mathbb R^3$ 可以是所谓的锯齿形区域，其边界由被二维球面截断的一维曲线片段组成。在适当的几何假设下，例如在 $Ω$ 和 $\partial Ω$ 上存在具有适当尺寸条件的加倍测度，我们构造了一类适应于该几何的退化椭圆算子 $L$，并建立了与这些算子相关的椭圆理论的关键估计。这包括边界 Poincaré 不等式和 Harnack 不等式、最大值原理以及解在边界处的 Hölder 连续性。我们引入了与该几何自然关联的 Hilbert 空间，构造了适当的迹算子和延拓算子，并用它们来定义 $Lu=0$ 的弱解。然后我们在 $Ω$ 内部和边界上证明 De Giorgi-Nash-Moser 估计，求解 Dirichlet 问题，从而构造出与 $L$ 相关联的椭圆测度 $ω_L$。最后，我们引入 Green 函数，并用它们证明了一个比较原理。由于我们的理论强调测度而非几何本身，因此即使在经典情形——半平面 $\mathbb R^2_+$ 的边界 $\partial \mathbb R^2_+= \mathbb R$ 上配备一个关于 $\mathbb R$ 上 Lebesgue 测度奇异的加倍测度 $μ$ 时，这些结果也是新的。最后，本文提供了著名的 Caffarelli-Sylvestre 延拓算子从其经典设定 $\mathbb R^{n+1}_+$ 到一般开集的推广，从而将分数阶 Laplacian 的概念推广到 Ahlfors 正则边界及其更一般的情形。

英文摘要：

Take an open domain $Ω\subset \mathbb R^n$ whose boundary may be composed of pieces of different dimensions. For instance, $Ω$ can be a ball on $\mathbb R^3$, minus one of its diameters $D$, or $Ω\subset \mathbb R^3$ could be a so-called saw-tooth domain, with a boundary consisting of pieces of 1-dimensional curves intercepted by 2-dimensional spheres. Under appropriate geometric assumptions, such as the existence of doubling measures on $Ω$ and $\partial Ω$ with appropriate size conditions, we construct a class of degenerate elliptic operators $L$ adapted to the geometry, and establish key estimates of elliptic theory associated to those operators. This includes boundary Poincaré and Harnack inequalities, maximum principle, and Hölder continuity of solutions at the boundary. We introduce Hilbert spaces naturally associated to the geometry, construct appropriate trace and extension operators, and use them to define weak solutions to $Lu=0$. Then we prove De Giorgi-Nash-Moser estimates inside $Ω$ and on the boundary, solve the Dirichlet problem and thus construct an elliptic measure $ω_L$ associated to $L$. At last, we introduce Green functions, and use them to prove a comparison principle. Since our theory emphasizes measures, rather than the geometry per se, the results are new even in the classical setting of a half-plane $\mathbb R^2_+$ when the boundary $\partial \mathbb R^2_+= \mathbb R$ is equipped with a doubling measure $μ$ singular with respect to the Lebesgue measure on $\mathbb R$. Finally, the present paper provides a generalization of the celebrated Caffarelli-Sylvestre extension operator from its classical setting of $\mathbb R^{n+1}_+$ to general open sets, and hence, an extension of the concept of fractional Laplacian to Ahlfors regular boundaries and beyond.

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