发表机构
ENSTA, Institut Polytechnique de Paris; University of Connecticut; University of Rochester（ENSTA，巴黎理工学院; 康涅狄格大学; 罗切斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将经典向量值迹理论推广至含分形边界的非Lipschitz $H^1$-延拓区域，定义切向与法向迹算子，建立相关空间的希尔伯特结构，推广麦克斯韦方程组的强制性双线性形式方法，可求解特定算子的边值问题。
AI 中文摘要
我们将经典的向量值切向与法向迹理论从Lipschitz区域推广到非Lipschitz的$H^1$-延拓区域的情形，这类区域包含具有分形边界的区域，如科赫雪花。我们基于$H^1(\boldsymbol{\u03a9})$中元素的迹算子的满射性以及相关迹空间的希尔伯特结构，对这些算子进行定义和研究。法向迹算子定义在$\boldsymbol{\u0399}^n$中的$\boldsymbol{\u0399div}$上。广义斯托克斯公式使我们能在二维和三维空间中，于$\boldsymbol{\u0399curl}$和$\boldsymbol{\u0399}^1(\boldsymbol{\u03a9})$上引入切向迹算子。遵循Buffa、Costabel和Sheen（2002）针对Lipschitz区域的方法，我们定义两个抽象切向边界空间作为这些迹算子的像，建立它们的希尔伯特结构，并构造一个抽象旋转算子来连接这两个空间，以替代基于法向量的几何旋转。我们还将Costabel（1991）针对麦克斯韦方程组的强制性双线性形式方法推广到非Lipschitz区域，得到的理论支持霍奇-狄拉克算子和格林公式的处理，并能在$H^1$-延拓区域框架内求解$\boldsymbol{\u03c1otv \u03c1otv +1}$算子的边值问题。
英文摘要
We generalize the classical vector-valued tangential and normal trace theory on Lipschitz domains to the setting of non-Lipschitz $H^1$-extension domains, which includes domains with fractal boundaries such as the Koch snowflake. We define and study these operators based on the surjectivity of the trace operator for elements of $H^1(Ω)$ and the Hilbert structure of the associated trace space. The normal trace operator is defined on $\boldsymbol{H}({\rm div},Ω)$ in $\mathbb{R}^n$. A generalized Stokes formula allows us to introduce the tangential trace operator on $\boldsymbol{H}({\bf curl},Ω)$ and $\boldsymbol{H}^1(Ω)$ in two and three dimensions. Following the approach of Buffa, Costabel and Sheen (2002) for Lipschitz domains, we define two abstract tangential boundary spaces as images of these trace operators, establish their Hilbert structure, and construct an abstract rotation operator linking them, in place of the geometric rotation based on the normal vector. We also extend Costabel's (1991) coercive bilinear form approach for Maxwell's equations to non-Lipschitz domains, yielding a theory that supports the treatment of the Hodge-Dirac operator and Green's formulas and enables the solution of boundary-value problems for the $\operatorname{\textbf{curl}}\operatorname{\textbf{curl}}+\textbf{I}$ operator within the $H^1$-extension domains framework.
CommentsVersion identique ; correction de la liste des fichiers source pour le transfert arXiv
