论文标题
具有混合维度边界的域中的椭圆理论
Elliptic theory in domains with boundaries of mixed dimension
论文作者
论文摘要
取一个开放的域$ω\ subset \ mathbb r^n $，其边界可能由不同的尺寸组成。例如，$ω$可以是$ \ mathbb r^3 $上的一个球，减去其直径之一$ d $，或$ω\ subset \ mathbb r^3 $可以是一个所谓的锯齿状域，其边界由2级固定器截获的1维曲线组成。在适当的几何假设下，例如在$ω$和$ \ $ \ partialω$上的双倍措施的情况下，我们构建了适合几何形状的退化椭圆运算符$ l $ $ l $，并建立与这些操作员相关的椭圆理论的关键估计。这包括边界庞加莱和哈纳克不平等，最大原则以及边界解决方案的霍德连续性。我们介绍了与几何形状自然相关的希尔伯特空间，构建适当的跟踪和扩展运算符，并使用它们将弱解决方案定义为$ lu = 0 $。然后，我们证明了$ω$内部的de Giorgi-nash-Moser估计，在边界上，解决了Dirichlet问题，从而构建了与$ L $相关的椭圆度量$ω_l$。最后，我们介绍了绿色功能，并使用它们来证明比较原则。由于我们的理论强调了衡量标准，而不是几何本质，因此即使在边界$ \ partial $ \ partial \ partbb r^2 _+= \ mathbb r $的经典环境中，结果也是新的，即使在半平面$ \ mathbb r^2 _+$中，结果也是新的。最后，本文从其经典的$ \ mathbb r^{n+1} _+$的经典设置到一般的开放式套件，从其经典的环境到一般的开放式套件，从而将著名的Caffarelli-Sylvestre扩展运算符提供了概括，因此，将分数Laplacian的概念扩展到ahlfors juarlfors juroparies and Beyond。
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.
