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

Abstract We obtain new estimates of stability for the continuation of solutions of elliptic inequalities <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mover accent="true"> <m:mo>∂</m:mo> <m:mo stretchy="false">¯</m:mo> </m:mover> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>⩽</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:math> |\bar{\partial}u|^{2}\leqslant a|u|^{2} from finite discrete sets. For this, we use Hörmander’s type of Carleman estimates for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover accent="true"> <m:mo>∂</m:mo> <m:mo stretchy="false">¯</m:mo> </m:mover> </m:math> {\bar{\partial}} with boundary terms and a special weight function equal to the linear combination 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>S</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {S_{n}} but we also introduce a new sequence of numbers <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>M</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {M_{n}} which gives us better estimates in the case of a finite number of observations.