The <i>L</i> ² -unique continuation property on manifolds with bounded geometry and the deformation operator

Abstract By definition, a differential operator T on a connected manifold satisfies the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {L^{2}} -unique continuation property if every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {L^{2}} -solution of T that vanishes on an open subset vanishes identically. We study the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {L^{2}} -unique continuation property of an operator T acting on a manifold with bounded geometry. In particular, we establish some connections between this property and the regularity properties of T . As an application, we prove that the deformation operator on a manifold with bounded geometry satisfies regularity and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {L^{2}} -unique continuation properties. As another application, we prove that suitable elliptic operators are invertible (Hadamard well-posedness). Our results apply, in particular, to compact manifolds, as all such manifolds have bounded geometry.