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We consider heat operators on a bounded domain $\Omega \subseteq\mathbb{R}^n$, with a critically singular potential diverging as the inversesquare of the distance to $\partial \Omega$. While null boundarycontrollability for such operators was recently proved in all dimensions inarXiv:2112.04457, it crucially assumed $\Omega$ was convex, and the controlmust be prescribed along the entire boundary. In this article, we prove insteadan approximate boundary controllability result for these operators, but we bothremove the convexity assumption on $\Omega$ and allow for the control to belocalized near a point $x_0 \in \partial \Omega$. In addition, we lower theregularity required on $\partial \Omega$ and the lower-order coefficients. Thekey novelty is a local Carleman estimate near $x_0$, with a carefully chosenweight that takes into account both the appropriate boundary conditions and thelocal geometry of $\partial \Omega$.

Approximate boundary controllability for parabolic equations with inverse square infinite potential wells