Theorem 1.If Dhas 14afa" title="Click to view the MathML source">P-property, then Dis a ball.
Four equivalent formulations of the Pompeiu problem are discussed.
A domain D has 14afa" title="Click to view the MathML source">P-property if there exists an f≠0, such that ∫Df(gx+y)dx=0 for all y∈R3 and all g∈SO(2), where a2290d3e98" 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 (∇2+k2)u=0in D, u∣S=1, 14a7a06c281d58a5e4f" title="Click to view the MathML source">uN∣S=0, and k>0is a constant, then Dis a ball.