Continuation of solutions of elliptic systems from discrete sets with application to geometry of surfaces

Abstract

We obtain new estimates of stability for continuation of solutions of elliptic inequali- ties | ∂u|2 < = a|u|2 from finite discrete sets. For this we use Hormander's type of Carleman estimates for ∂-bar with boundary terms and a special weight function equal to the linear com-bination of Green's functions with singularities at points where our function is given. To estimate functions in a given domain we use not only the classical Blaschke partial sum Sn but we also introduce a new sequence of numbers Mn which gives us better estimates in the case of a finite number of observations.