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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si1.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=9b60f2382aaa7af02015bc8091873072" title="Click to view the MathML source">D⊂R³class="mathContainer hidden">class="mathCode">$D\subset {\mathbb{R}}^3$ is a bounded domain with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si2.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=a77e2912385ce45f3859068a7171b616" title="Click to view the MathML source">C¹class="mathContainer hidden">class="mathCode">$C^1$-smooth boundary. Our result is:

class="boldFont">Theorem 1.If  class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si3.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=759b670fd02ad6fbfdf850c8dffbb43e" title="Click to view the MathML source">Dclass="mathContainer hidden">class="mathCode">$D$has  class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si4.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=a9296249511d5c96b9f1990863714afa" title="Click to view the MathML source">Pclass="mathContainer hidden">class="mathCode">$P$-property, then  class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si3.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=759b670fd02ad6fbfdf850c8dffbb43e" title="Click to view the MathML source">Dclass="mathContainer hidden">class="mathCode">$D$is a ball.

Four equivalent formulations of the Pompeiu problem are discussed.

A domain class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si3.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=759b670fd02ad6fbfdf850c8dffbb43e" title="Click to view the MathML source">Dclass="mathContainer hidden">class="mathCode">$D$ has class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si4.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=a9296249511d5c96b9f1990863714afa" title="Click to view the MathML source">Pclass="mathContainer hidden">class="mathCode">$P$-property if there exists an class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si8.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=d4d9918164982e3b1811bbf40f7e17da" title="Click to view the MathML source">f≠0class="mathContainer hidden">class="mathCode">$f\ne 0$, class="mathmlsrc">title="View 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">class="imgLazyJSB inlineImage" height="18" width="77" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0893965916302099-si9.gif">class="mathContainer hidden">class="mathCode">$f\in L_{loc}^1\left({\mathbb{R}}^3\right)$ such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si10.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=2ecbbae53155cf120a1953484bdad219" title="Click to view the MathML source">∫_Df(gx+y)dx=0class="mathContainer hidden">class="mathCode">${\int }_{D}f(gx+y)dx=0$ for all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si11.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=783081f9cabc875b3c793387e3f89ebd" title="Click to view the MathML source">y∈R³class="mathContainer hidden">class="mathCode">$y\in {\mathbb{R}}^3$ and all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si12.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=9b5536035115afd57fa9bd3fa376718c" title="Click to view the MathML source">g∈SO(2)class="mathContainer hidden">class="mathCode">$g\in SO\left(2\right)$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si13.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=aa884a7f1b29a6675da438a2290d3e98" title="Click to view the MathML source">SO(2)class="mathContainer hidden">class="mathCode">$SO\left(2\right)$ is the rotation group.

The result obtained concerning the related symmetry problem is:

class="boldFont">Theorem 2.If  class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si14.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=45b5868e95f9e764e0ec96deecfd6e21" title="Click to view the MathML source">(∇²+k²)u=0class="mathContainer hidden">class="mathCode">$({\nabla }^2+k^2)u=0$in  class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si3.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=759b670fd02ad6fbfdf850c8dffbb43e" title="Click to view the MathML source">Dclass="mathContainer hidden">class="mathCode">$D$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si16.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=a03ab6f81914b0e5ec97c0ea404304bb" title="Click to view the MathML source">u∣_S=1class="mathContainer hidden">class="mathCode">$u{\mid }_S=1$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si17.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=5114279c917d214a7a06c281d58a5e4f" title="Click to view the MathML source">u_N∣_S=0class="mathContainer hidden">class="mathCode">$u_N{\mid }_S=0$, and  class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si18.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=7493ddd2a1d104f249f654e22d23e8ab" title="Click to view the MathML source">k>0class="mathContainer hidden">class="mathCode">$k>0$is a constant, then  class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302099&_mathId=si3.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=759b670fd02ad6fbfdf850c8dffbb43e" title="Click to view the MathML source">Dclass="mathContainer hidden">class="mathCode">$D$is a ball.