A problem in analysis

Abstract

Assume that D R 2 is a bounded domain, diffeomorphic to a disc, star-shaped, with a C 1; boundary C , > 0, which can be represented in polar coordinates as r D f . /, where f > 0 is a smooth 2 -periodic function. Let ˙n WD ˙n . / WD e˙i n f n . /. Theorem. Assume that Z 2 0 ˙n f 2 . /d D 0 nD 1;2;:::Then f D const.