| Abstract: |
| This paper investigates the well-posedness and approximation of inverse problems formulated on highly irregular spatial domains, including those with fractal boundaries. The primary objective is the identification of a constant diffusivity parameter, $\alpha_* > 0$, within an abstract parabolic Cauchy problem. This identification relies on a quadratic overdetermining condition, which requires measuring the $L^2$-norm of the solution`s energy at a fixed measuring time-instant $T > 0$.
To address the geometric complexities inherent to irregular domains, we introduce a sequence of approximating direct and inverse problems defined on smoother, more regular spatial domains. The rigorous convergence of these approximations to the exact solution of the original inverse problem is established through the Mosco convergence of the associated Dirichlet energy forms. Crucially, appropriate scaling is fundamental in proving this Mosco convergence of the energies, precisely because there is a jump of dimension when passing from the regular approximating domains to the highly irregular or fractal limit geometry. Finally, we will also address the case of a fractional-in-time inverse problem, demonstrating the versatility of the proposed mathematical framework. |
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