Realization of the fractional Laplacian with nonlocal exterior conditions via forms method

Claus, Burkhard; Warma, Mahamadi

doi:10.48550/arxiv.1904.13312

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2019

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“…Notice that (−∆) s D is the realization in L 2 (Ω) of the fractional Laplace operator (−∆) s with the Dirichlet exterior condition u = 0 in R N \ Ω. We refer to [19] for a rigorous definition of (−∆) s D . Finally, we close this section by recalling the integration-by-parts formula for (−∆) s (see e.g.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Optimal Control of Fractional Elliptic PDEs with State Constraints and Characterization of the dual of Fractional Order Sobolev Spaces

Antil¹,

Verma²,

Warma³

2019

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This paper introduces the notion of state constraints for optimal control problems governed by fractional elliptic PDEs of order s ∈ (0, 1). There are several mathematical tools that are developed during the process to study this problem, for instance, the characterization of the dual of the fractional order Sobolev spaces and well-posedness of fractional PDEs with measure-valued datum. These tools are widely applicable. We show well-posedness of the optimal control problem and derive the first order optimality conditions. Notice that the adjoint equation is a fractional PDE with measure as the right-hand-side datum. We use the characterization of the fractional order dual spaces to study the regularity of the state and adjoint equations. We emphasize that the classical case (s = 1) was considered by E. Casas in [14] but almost none of the existing results are applicable to our fractional case.

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Section: Notation and Preliminariesmentioning

confidence: 99%