Solution to the Pompeiu problem and the related symmetry problem

详细信息    查看全文

• 作者：A.G. Ramm ^{ramm@math.ksu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：The Pompeiu problem ; Fourier transforms of characteristic sets ; Over-determined boundary value problems ; Symmetry problems
• 刊名：Applied Mathematics Letters
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：63
• 期：Complete
• 页码：28-33
• 全文大小：626 K

文摘

Assume that bc8091873072" title="Click to view the MathML source">D⊂R³$D\subset {\mathbb{R}}^3$ is a bounded domain with C¹$C^1$-smooth boundary. Our result is:

Theorem 1.If  D$D$has  P$P$-property, then  D$D$is a ball.

Four equivalent formulations of the Pompeiu problem are discussed.

A domain D$D$ has P$P$-property if there exists an f≠0$f\ne 0$, $f\in L_{loc}^1\left({\mathbb{R}}^3\right)$ such that bae53155cf120a1953484bdad219" title="Click to view the MathML source">∫_Df(gx+y)dx=0${\int }_{D}f(gx+y)dx=0$ for all bc875b3c793387e3f89ebd" title="Click to view the MathML source">y∈R³$y\in {\mathbb{R}}^3$ and all g∈SO(2)$g\in SO\left(2\right)$, where e98" title="Click to view the MathML source">SO(2)$SO\left(2\right)$ is the rotation group.

The result obtained concerning the related symmetry problem is:

Theorem 2.If  e95f9e764e0ec96deecfd6e21" title="Click to view the MathML source">(∇²+k²)u=0$({\nabla }^2+k^2)u=0$in  D$D$, a03ab6f81914b0e5ec97c0ea404304bb" title="Click to view the MathML source">u∣_S=1$u{\mid }_S=1$, a06c281d58a5e4f" title="Click to view the MathML source">u_N∣_S=0$u_N{\mid }_S=0$, and  a1d104f249f654e22d23e8ab" title="Click to view the MathML source">k>0$k>0$is a constant, then  D$D$is a ball.