Fractional Elliptic Problems on Lipschitz Domains: Regularity and Approximation

Motivated by many applications, optimal control problems with integer controls have recently received a significant attention. Some state-of-the-art work uses perimeter-regularization to derive stationarity conditions and trust-region algorithms. However, the discretization is difficult in this case because the perimeter is concentrated on a set of dimension $$d - 1$$ d - 1 for a domain of dimension d. This article proposes a potential way to overcome this challenge by using the fractional nonlocal perimeter with fractional exponent $$0<\alpha <1$$ 0 < α < 1 . In this way, the boundary integrals in the perimeter regularization are replaced by volume integrals. Besides establishing some non-trivial properties associated with this perimeter, a $$\Gamma $$ Γ -convergence result is derived. This result establishes convergence of minimizers of fractional perimeter-regularized problem, to the standard one, as the exponent $$\alpha $$ α tends to 1. In addition, the stationarity results are derived and algorithmic convergence analysis is carried out for $$\alpha \in (0.5,1)$$ α ∈ ( 0.5 , 1 ) under an additional assumption on the gradient of the reduced objective. The theoretical results are supplemented by a preliminary computational experiment. We observe that the isotropy of the total variation may be approximated by means of the fractional perimeter functional.

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Fractional Elliptic Problems on Lipschitz Domains: Regularity and Approximation

Cited by 2 publications

References 93 publications

Higher-order asymptotic expansions and finite difference schemes for the fractional p-Laplacian

Higher-order asymptotic expansions and finite difference schemes for the fractional p-Laplacian

Integer Optimal Control with Fractional Perimeter Regularization

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