Theorem 1.If bb43e" title="Click to view the MathML source">Dhas P-property, then bb43e" title="Click to view the MathML source">Dis a ball.
Four equivalent formulations of the Pompeiu problem are discussed.
A domain bb43e" title="Click to view the MathML source">D has P-property if there exists an bbf40f7e17da" title="Click to view the MathML source">f≠0, 9c5"> such that bbae53155cf120a1953484bdad219" title="Click to view the MathML source">∫Df(gx+y)dx=0 for all 9cabc875b3c793387e3f89ebd" title="Click to view the MathML source">y∈R3 and all g∈SO(2), where 84a7f1b29a6675da438a2290d3e98" 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 e95f9e764e0ec96deecfd6e21" title="Click to view the MathML source">(∇2+k2)u=0in bb43e" title="Click to view the MathML source">D, bb" title="Click to view the MathML source">u∣S=1, 9c917d214a7a06c281d58a5e4f" title="Click to view the MathML source">uN∣S=0, and k>0is a constant, then bb43e" title="Click to view the MathML source">Dis a ball.