Carleman commutator approach in logarithmic convexity for parabolic equations

TL;DR

This paper introduces a novel approach combining logarithmic convexity and Carleman commutators to derive observation estimates and spectral inequalities for heat equations, including those with inverse square potentials.

Contribution

The paper presents a new method integrating logarithmic convexity and Carleman commutators for heat equations, extending to inverse square potentials and deriving spectral inequalities.

Findings

Established an observation estimate at one time for the heat equation.

Derived a spectral inequality for the eigenvalue problem with inverse square potential.

Extended the approach to heat equations with singular potentials.

Abstract

In this paper we investigate on a new strategy combining the logarithmic convexity (or frequency function) and the Carleman commutator to obtain an observation estimate at one time for the heat equation in a bounded domain. We also consider the heat equation with an inverse square potential. Moreover, a spectral inequality for the associated eigenvalue problem is derived.