庞培问题及相关对称性问题的解
Solution to the Pompeiu problem and the related symmetry problem
摘要：
假设$D\subset \mathbb{R}^3$是一个有界区域，具有$C^1-$光滑边界。我们的结论是：{ \bf定理1}。{ \em如果$D$具有$P-$属性，则$D$是球。} Pompeiu问题的四个等价公式进行了讨论。如果存在一个$f\neq 0$，$f\in L^1_{loc}(\mathbb{R}^3)$，使得所有$y\in \mathbb{R}^3$和所有$g\in SO(2)$的$\int_{D}f(gx+y)dx=0$，其中$ SO(2)$是旋转群，则域$D$具有$P-$属性。关于相关的对称性问题得到的结果是：{ \bf定理2}。{ \em如果$D$、$u|_S=1$、$u_N|_S=0$和$k>0$中的$(\nabla^2 +k^2)u=0$是常数，则$D$是球。}
英文摘要：
Assume that $D\subset \mathbb{R}^3$ is a bounded domain with $C^1-$smooth boundary. Our result is: {\bf Theorem 1.} {\em If $D$ has $P-$property, then $D$ is a ball.} Four equivalent formulations of the Pompeiu problem are discussed. A domain $D$ has $P-$property if there exists an $f\neq 0$, $f\in L^1_{loc}(\mathbb{R}^3)$ such that $\int_{D}f(gx+y)dx=0$ for all $y\in \mathbb{R}^3$ and all $g\in SO(2)$, where $ SO(2)$ is the rotation group. The result obtained concerning the related symmetry problem is: {\bf Theorem 2.} {\em If $(\nabla^2 +k^2)u=0$ in $D$, $u|_S=1$, $u_N|_S=0$, and $k>0$ is a constant, then $D$ is a ball.}
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。
