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In [Antil et al. Inverse Probl. 35 (2019) 084003.] we introduced a new notion of optimal control and source identification (inverse) problems where we allow the control/source to be outside the domain where the fractional elliptic PDE is fulfilled. The current work extends this previous work to the parabolic case. Several new mathematical tools have been developed to handle the parabolic problem. We tackle the Dirichlet, Neumann and Robin cases. The need for these novel optimal control concepts stems from the fact that the classical PDE models only allow placing the control/source either on the boundary or in the interior where the PDE is satisfied. However, the nonlocal behavior of the fractional operator now allows placing the control/source in the exterior. We introduce the notions of weak and very-weak solutions to the fractional parabolic Dirichlet problem. We present an approach on how to approximate the fractional parabolic Dirichlet solutions by the fractional parabolic Robin solutions (with convergence rates). A complete analysis for the Dirichlet and Robin optimal control problems has been discussed. The numerical examples confirm our theoretical findings and further illustrate the potential benefits of nonlocal models over the local ones.
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This content will become publicly available on January 1, 2027
Behavior of absorbing and generating p -Robin eigenvalues in bounded and exterior domains
We establish rigorous quantitative inequalities for the first eigenvalue of the generalized π-Robin problem, for both the classical diffusion absorption case, where the Robin boundary parameter πΌ is positive, and the superconducting generation regime (πΌ < 0), where the boundary acts as a source. In bounded domains, we use a unified approach to derive a precise asymptotic behavior for all π and all small real πΌ, improving existing results in various directions, including requiring weaker boundary regularity for the case of the classical 2-Robin problem, studied in the fundamental work by RenΓ© Sperb. In exterior domains, we characterize the existence of eigenvalues, establish general inequalities and asymptotics as πΌ β 0 for the first eigenvalue of the exterior of a ball, and obtain some sharp geometric inequalities for convex domains in two dimensions.
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- Award ID(s):
- 2345500
- PAR ID:
- 10650602
- Publisher / Repository:
- ELSEVIER
- Date Published:
- Journal Name:
- Nonlinear Analysis
- Volume:
- 262
- Issue:
- C
- ISSN:
- 0362-546X
- Page Range / eLocation ID:
- 113943
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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