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 authorsup>
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 <span 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⊂R3sup>span>

si1.gif" overflow="

scroll">D⊂sup>struck">R3sup>span>span>span> is a bounded domain with <span 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">C1sup>span>

si2.gif" overflow="

scroll">sup>C1sup>span>span>span>-smooth boundary. Our result is:

sp000010">Theorem 1.strong>If <span 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">Dspan> $si3.gif" overflow="$ scroll">Dspan>span>span>has <span 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">Pspan> $si4.gif" overflow="$ scroll">Pspan>span>span>-property, then <span 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">Dspan> $si3.gif" overflow="$ scroll">Dspan>span>span>is a ball.

sp000015">Four equivalent formulations of the Pompeiu problem are discussed.

sp000020">A domain <span 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">Dspan> $si3.gif" overflow="$ scroll">Dspan>span>span> has <span 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">Pspan> $si4.gif" overflow="$ scroll">Pspan>span>span>-property if there exists an <span 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≠0span> $si8.gif" overflow="$ scroll">f≠0span>span>span>, 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">ss="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">script>style="vertical-align:bottom" width="77" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0893965916302099-si9.gif">script> $si9.gif" overflow="$ scroll">f∈subsup>Lloc1subsup>(sup>struck">R3sup>)span>span>span> such that <span 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">∫Dsub>f(gx+y)dx=0span> $si10.gif" overflow="$ scroll">sub>∫Dsub>f(gx+y)dx=0span>span>span> for all <span 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∈R3sup>span> $si11.gif" overflow="$ scroll">y∈sup>struck">R3sup>span>span>span> and all <span 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)span> $si12.gif" overflow="$ scroll">g∈SO(2)span>span>span>, where <span 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)span> $si13.gif" overflow="$ scroll">SO(2)span>span>span> is the rotation group.

sp000025">The result obtained concerning the related symmetry problem is:

sp000030">Theorem 2.strong>If <span 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">(∇2sup>+k2sup>)u=0span> $si14.gif" overflow="$ scroll">(sup>∇2sup>+sup>k2sup>)u=0span>span>span>in <span 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">Dspan> $si3.gif" overflow="$ scroll">Dspan>span>span>, <span 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∣Ssub>=1span> $si16.gif" overflow="$ scroll">usub>∣Ssub>=1span>span>span>, <span 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">uNsub>∣Ssub>=0span> $si17.gif" overflow="$ scroll">sub>uNsub>sub>∣Ssub>=0span>span>span>, and <span 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>0span> $si18.gif" overflow="$ scroll">k>0span>span>span>is a constant, then <span 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">Dspan> $si3.gif" overflow="$ scroll">Dspan>span>span>is a ball.

地址：北京市海淀区学院路29号邮编：100083

电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700