54 research outputs found
Unique Continuation for Sublinear Elliptic Equations Based on Carleman Estimates
In this article we deal with different forms of the unique continuation
property for second order elliptic equations with nonlinear potentials of
sublinear growth. Under suitable regularity assumptions, we prove the weak and
the strong unique continuation property. Moreover, we also discuss the unique
continuation property from measurable sets, which shows that nodal domains to
these equations must have vanishing Lebesgue measure. Our methods rely on
suitable Carleman estimates, for which we include the sublinear potential into
the main part of the operator.Comment: 22 page
Quantitative Runge Approximation and Inverse Problems
In this short note we provide a quantitative version of the classical Runge
approximation property for second order elliptic operators. This relies on
quantitative unique continuation results and duality arguments. We show that
these estimates are essentially optimal. As a model application we provide a
new proof of the result from \cite{F07}, \cite{AK12} on stability for the
Calder\'on problem with local data.Comment: 12 page
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