Unique continuation for a class of higher order elliptic operators

Abstract

where P is an elliptic operator with constant coefficients and V is a singular potential. We will prove that if n > m, P is an elliptic operator of order m ≥ 2 whose principal part satisfies conditions that will be specified later, V ∈ L n m (R), and if U ∈ H(R), p = 2n m+n , satisfies (0.1), then U is identically zero if its support is contained in a half space whose normal direction satisfies a hypothesis involving the symbol of P . By H(R) we mean the space of functions with m derivatives in L(R). It is well known that the above unique continuation property, (u.c.p. henceforth in this paper), for the solutions of the differential Inequality (0.1), follows from the proof of a weighted inequality of the form