论文标题
Lipschitz Constant的Lipschitz域边界附近的椭圆pdes的零尺寸
Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant
论文作者
论文摘要
令$ω\ subset \ mathbb r^d $为$ c^1 $域，或者更一般而言，是一个带有小lipschitz常数的Lipschitz域，$ a（x）$是$ d \ times d $均匀的椭圆形，具有Lipschitz系数的对称矩阵。假设$ u $在$ω$中是谐音，或者具有更大的常规$ u $ solves $ \ permatatorName {div}（a（x）\ nabla u）= 0 $ in $ω$中的0 $，而$ u $在$σ= \partialΩ\partialΩ\partialΩ\partialΩ\partialΩ\ cap b $ for All $ b $中。我们研究了$σ$中$ u $的单数尺寸，特别是我们表明，有一个可数的开放球$（b_i）_i $的家族，使得$ | _ {b_i \capΩ} $不更改符号，而$ k \ backslash \ bickslash \ bigcup_i b_i b_i b_i b_i beg big big big big big d $ d-for $ d-for $ d-1- $ d-1-- σ$。我们还找到了$（D-1）$ - 尺寸的hausdorff的上限，$ u u $ in Balls的零集与频率相交的$σ$相交。 结果，我们证明了这类函数的边界上的一个新的独特延续原理，并表明在所有点的所有点上消失的顺序是有限的，除了一组Hausdorff Dimension，最多$ d-1-ε$。
Let $Ω\subset \mathbb R^d$ be a $C^1$ domain or, more generally, a Lipschitz domain with small Lipschitz constant and $A(x)$ be a $d \times d$ uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume $u$ is harmonic in $Ω$, or with greater generality $u$ solves $\operatorname{div}(A(x)\nabla u)=0$ in $Ω$, and $u$ vanishes on $Σ= \partialΩ\cap B$ for some ball $B$. We study the dimension of the singular set of $u$ in $Σ$, in particular we show that there is a countable family of open balls $(B_i)_i$ such that $u|_{B_i \cap Ω}$ does not change sign and $K \backslash \bigcup_i B_i$ has Minkowski dimension smaller than $d-1-ε$ for any compact $K \subset Σ$. We also find upper bounds for the $(d-1)$-dimensional Hausdorff measure of the zero set of $u$ in balls intersecting $Σ$ in terms of the frequency. As a consequence, we prove a new unique continuation principle at the boundary for this class of functions and show that the order of vanishing at all points of $Σ$ is bounded except for a set of Hausdorff dimension at most $d-1-ε$.
