Solution to the Pompeiu problem and the related symmetry problem

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• 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 the MathML source">D⊂R³$D\subset {\mathbb{R}}^3$ is a bounded domain with the MathML source">C¹$C^1$-smooth boundary. Our result is:

Theorem 1.If  the MathML source">D$D$has  the MathML source">P$P$-property, then  the MathML source">D$D$is a ball.

Four equivalent formulations of the Pompeiu problem are discussed.

A domain the MathML source">D$D$ has the MathML source">P$P$-property if there exists an the MathML source">f≠0$f\ne 0$, the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si9.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=0c081304b01d7cef1b163e768979b9c5">the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0893965916302099-si9.gif">$f\in L_{loc}^1\left({\mathbb{R}}^3\right)$ such that the MathML source">∫_Df(gx+y)dx=0${\int }_{D}f(gx+y)dx=0$ for all the MathML source">y∈R³$y\in {\mathbb{R}}^3$ and all the MathML source">g∈SO(2)$g\in SO\left(2\right)$, where 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  the MathML source">(∇²+k²)u=0$({\nabla }^2+k^2)u=0$in  the MathML source">D$D$, the MathML source">u∣_S=1$u{\mid }_S=1$, the MathML source">u_N∣_S=0$u_N{\mid }_S=0$, and  the MathML source">k>0$k>0$is a constant, then  the MathML source">D$D$is a ball.