Unique continuation for |?u| = V |Ñu| and related problems.

Much of this paper will be concerned with the proof of the following Theorem 1. Suppose d = 3, r = max {d, (3d - 4)/2}. If V Î Llocr(Rd), then the differential inequality |?u| = V |Ñu| has the strong unique continuation property in the following sense: If u belongs to the Sobolev space Wloc2,p and if |?u| = V |Ñu| and limR?0 R-N ?|x| < R |Ñu|p' = 0 for all N then u is constant