2023/11/02 by Shao, Arick, Vergara, Bruno · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2311.01628
We consider heat operators on a bounded domain Ω⊆ ℝn, with a critically singular potential diverging as the inverse square of the distance to ∂ Ω. While null boundary controllability for such operators was recently proved in all dimensions in arXiv:2112.04457, it crucially assumed (i) Ω was convex, (ii) the control must be prescribed along all of ∂ Ω, and (iii) the strength of the singular potential must be restricted to a particular subrange. In this article, we prove instead a definitive approximate boundary control result for these operators, in that we (i) do not assume convexity of Ω, (ii) allow for the control to be localized near any x0 ∈ ∂ Ω, and (iii) treat the full range of strength parameters for the singular potential. Morever, we lower the regularity required for ∂ Ω and the lower-order coefficients. The key novelty is a local Carleman estimate near x0, with a carefully chosen weight that takes into account both the appropriate boundary conditions and the local geometry of ∂ Ω.