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University of Birmingham > Talks@bham > Analysis Seminar > The Kato Square Root Problem for Divergence Form Operators with Potential
The Kato Square Root Problem for Divergence Form Operators with PotentialAdd to your list(s) Download to your calendar using vCal
If you have a question about this talk, please contact Andrew Morris. The Kato square root problem for divergence form elliptic operators is the equivalence statement $\left\Vert \sqrt { \mathrm{div} \left( A \nabla \right)}u \right\Vert \simeq \left\Vert \nabla u \right\Vert$, where $A$ is a complex matrix-valued function. In 2006, a few years after the first proof of this statement, A. Axelsson, S. Keith and A. McIntosh developed a general framework for proving square function estimates associated with Dirac-type operators and they showed that the Kato problem followed as an immediate application. In this talk I will give an overview of the Kato square root problem and run through a sketch of the proof of Axelsson, Keith and McIntosh. I will then discuss a generalisation of the Kato problem to include positive potentials $V$, namely $\left\Vert \sqrt{-\mathrm{div} \left( A \nabla \right) + V}u \right\Vert \simeq \left\Vert \nabla u \right\Vert + \left\Vert V^{\frac{1}{2}} u \right\Vert$. I will discuss how the Axelsson-Keith-McIntosh framework can be altered to allow for dependence of the Dirac-type operator on the potential. The Kato estimate for certain potentials will then follow as a result. This talk is part of the Analysis Seminar series. This talk is included in these lists:Note that ex-directory lists are not shown. |
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