拟凸域上散度型椭圆方程的边界唯一延拓性
Unique continuation at the boundary for divergence form elliptic equations on quasiconvex domains
AI总结：
本文研究拟凸Lipschitz域上散度型椭圆方程的边界唯一延拓问题，证明存在可数开球族使解在各球与域的交集上保号，且紧子集与开球并的差集Minkowski维数严格小于d-1-ε，由此在拟凸域下证明了Lin猜想。
摘要：
设$Ω\subset \mathbb{R}^d$为拟凸Lipschitz域，$A(x)$为系数满足Lipschitz条件的$d \times d$一致椭圆对称矩阵。假设非平凡函数$u$在$Ω$内满足方程$-\nabla \cdot (A(x) \nabla u) = 0$，且在某个球$B$与$Ω$边界的交集$Σ= \partial Ω\cap B$上$u$取值为零。本文的主要贡献是证明存在一族可数开球$(B_i)_i$，使得$u$在每个$B_i \cap Ω$上的限制保持一致符号。此外，对于$Σ$的任意紧子集$K$，差集$K \setminus \bigcup_i B_i$的Minkowski维数严格小于$d - 1 - ε$。作为推论，我们在拟凸域情形下证明了Lin猜想。
英文摘要：
Let $Ω\subset \mathbb{R}^d$ be a quasiconvex Lipschitz domain and $A(x)$ be a $d \times d$ uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume a nontrivial $u$ solves $-\nabla \cdot (A(x) \nabla u) = 0$ in $Ω$, and $u$ vanishes on $Σ= \partial Ω\cap B$ for some ball $B$. The main contribution of this paper is to demonstrate the existence of a countable collection of open balls $(B_i)_i$ such that the restriction of $u$ to $B_i \cap Ω$ maintains a consistent sign. Furthermore, for any compact subset $K$ of $Σ$, the set difference $K \setminus \bigcup_i B_i$ is shown to possess a Minkowski dimension that is strictly less than $d - 1 - ε$. As a consequence, we prove Lin's conjecture in quasiconvex domains.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
