Unique Continuation for Sublinear Elliptic Equations Based on Carleman Estimates

TL;DR

This paper establishes weak and strong unique continuation properties for second order elliptic equations with sublinear nonlinear potentials using Carleman estimates, including from measurable sets, under regularity assumptions.

Contribution

It introduces novel Carleman estimates that incorporate sublinear potentials directly into the operator, enabling new unique continuation results for nonlinear elliptic equations.

Findings

Proved weak and strong unique continuation properties.

Established unique continuation from measurable sets.

Demonstrated that nodal domains have vanishing Lebesgue measure.

Abstract

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.