Strong Unique Continuation Properties for Elliptic Operators

Abstract

In this chapter, we introduce the notion of strong unique continuation for solutions of elliptic equations. Under suitable assumptions on the potential V, a solution of an elliptic differential inequality of type \(\vert {\Delta u}\vert \le \vert {Vu}\vert \) which is flat at a given point should vanish on the connected component of that point. We start with results involving critical radial potentials such as \(V(x)=\vert x\vert ^{-2}\), using essentially some \(L^{2}\) Carleman estimates due to R. Regbaoui. [117–119]. To handle the critical case \(V\in L^{n/2}\), we give an exposition of the D. Jerison and C. Kenig result [68] involving \(L^{p}-L^{q}\) Carleman estimates with singular weights, following a method introduced by C. Sogge. in [139]. The last sections of this chapter are concerned with T. Wolff’s measure-theoretic lemma used to modify Carleman’s method and to obtain some weak unique continuation results with critical potentials.