Carleman inequalities for the Dirac and Laplace operators and unique continuation (Q1092314)

The Schrödinger operator \((L+V)\), where L is the Laplacian and V a potential, satisfies the unique continuation property if solutions of \((L+V)u=0\) which vanish in a non-empty, open subset of a connected set vanish identically. The Schrödinger operator is known to have the unique continuation property provided certain so-called Carleman inequalities hold. Reference is made to the large literature on these inequalities and their applications to uniqueness questions. In this paper best possible Carleman-type inequalities are obtained for the case when L is replaced by the Dirac operator D. It is shown that \((D+V)\) has the unique continuation property provided \(V\in L^{\gamma}_{loc}({\mathbb{R}}^ n)\), \(\gamma =(3n-2)/2\), \(n\geq 3\) which offers an improvement on known results. An additional feature of the paper is that the approach adopted to the treatment of the Dirac operator enables a simpler proof to be obtained of best possible Carleman-type inequalities for the Laplace operator.