Theorem 1.If 302099&_mathId=si3.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=759b670fd02ad6fbfdf850c8dffbb43e" title="Click to view the MathML source">Dhas 302099&_mathId=si4.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=a9296249511d5c96b9f1990863714afa" title="Click to view the MathML source">P-property, then 302099&_mathId=si3.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=759b670fd02ad6fbfdf850c8dffbb43e" title="Click to view the MathML source">Dis a ball.
Four equivalent formulations of the Pompeiu problem are discussed.
A domain 302099&_mathId=si3.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=759b670fd02ad6fbfdf850c8dffbb43e" title="Click to view the MathML source">D has 302099&_mathId=si4.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=a9296249511d5c96b9f1990863714afa" title="Click to view the MathML source">P-property if there exists an 302099&_mathId=si8.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=d4d9918164982e3b1811bbf40f7e17da" title="Click to view the MathML source">f≠0, 302099&_mathId=si9.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=0c081304b01d7cef1b163e768979b9c5">302099-si9.gif"> such that 302099&_mathId=si10.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=2ecbbae53155cf120a1953484bdad219" title="Click to view the MathML source">∫Df(gx+y)dx=0 for all 302099&_mathId=si11.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=783081f9cabc875b3c793387e3f89ebd" title="Click to view the MathML source">y∈R3 and all 302099&_mathId=si12.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=9b5536035115afd57fa9bd3fa376718c" title="Click to view the MathML source">g∈SO(2), where 302099&_mathId=si13.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=aa884a7f1b29a6675da438a2290d3e98" title="Click to view the MathML source">SO(2) is the rotation group.
The result obtained concerning the related symmetry problem is:
Theorem 2.If 302099&_mathId=si14.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=45b5868e95f9e764e0ec96deecfd6e21" title="Click to view the MathML source">(∇2+k2)u=0in 302099&_mathId=si3.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=759b670fd02ad6fbfdf850c8dffbb43e" title="Click to view the MathML source">D, 302099&_mathId=si16.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=a03ab6f81914b0e5ec97c0ea404304bb" title="Click to view the MathML source">u∣S=1, 302099&_mathId=si17.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=5114279c917d214a7a06c281d58a5e4f" title="Click to view the MathML source">uN∣S=0, and 302099&_mathId=si18.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=7493ddd2a1d104f249f654e22d23e8ab" title="Click to view the MathML source">k>0is a constant, then 302099&_mathId=si3.gif&_user=111111111&_pii=S0893965916302099&_rdoc=1&_issn=08939659&md5=759b670fd02ad6fbfdf850c8dffbb43e" title="Click to view the MathML source">Dis a ball.