Keskeiset käsitteet
This study investigates the problem of identifying the unknown interior Robin boundary of a connected domain using Cauchy data from the exterior region of a harmonic function. It proposes two shape optimization formulations employing least-squares boundary-data-tracking cost functionals and establishes the existence of optimal shape solutions. The study also demonstrates the ill-posed nature of the shape optimization formulations and employs multiple sets of Cauchy data to address the difficulty of detecting concavities in the unknown boundary.
Tiivistelmä
The study revisits the problem of identifying the unknown interior Robin boundary of a connected domain using Cauchy data from the exterior region of a harmonic function. It investigates two shape optimization reformulations employing least-squares boundary-data-tracking cost functionals.
Key highlights:
- The study rigorously addresses the existence of optimal shape solutions, filling a gap in the literature.
- It demonstrates the ill-posed nature of the two shape optimization formulations by establishing the compactness of the Riesz operator associated with the quadratic shape Hessian corresponding to each cost functional.
- The study employs multiple sets of Cauchy data to address the difficulty of detecting concavities in the unknown boundary.
- Numerical experiments in two and three dimensions illustrate the numerical procedure relying on Sobolev gradients proposed herein.
Tilastot
The study does not provide any specific numerical data or metrics to support the key logics. It focuses on the theoretical analysis and numerical investigation of the problem.
Lainaukset
There are no direct quotes from the content that support the key logics.