Unique Continuation from Sets of Positive Measure

Abstract

Let Ω be a connected open subset of ℝ n and let V, W be functions on Ω. We say that the differential inequality $$\left| {\Delta u} \right| \leqslant \left| {Vu} \right| + \left| {W\nabla u} \right|$$ (1.1) has the weak unique continuation property (w.u.c.p) if any solution u of (1.1) which vanishes on an open subset of Ω is identically zero. And we say that (1.1) has the strong unique continuation property (s.u.c.p) if any solution u is identically zero whenever it vanishes of infinite order at a point of Ω. We recall that a function \(u \in L_{{loc}}^{p}\) is said to vanish of infinite order at a point x 0 (or that u is flat at x 0) if for all N > 0, $$\int_{{\left| {x - {{x}_{0}}} \right|}} {{{{\left| {u\left( x \right)} \right|}}^{p}}dx = O\left( {{{R}^{N}}} \right)asR \to 0} .$$ (1.2)