Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation

We consider the quasi-reversibility method to solve the Cauchy problem for Laplace's equation in a smooth bounded domain. We assume that the Cauchy data are contaminated by some noise of amplitude or, so that we make a regular choice of e as a function of or, where e is the small parameter of the quasi-reversibility method. Specifically, we present two different results concerning the convergence rate of the solution of quasi-reversibility to the exact solution when or tends to 0. The first result is a convergence rate of type 1/(log1/sigma)(beta) in a truncated domain, the second one holds when a source condition is assumed and is a convergence rate of type sigma 1/2 in the whole domain.