We study a class of integral functionals known as nonlocal perimeters, which, intuitively, express a weighted interaction between a set and its complement. The weight is provided by a positive kernel K, which might be singular.In the first part of the paper, we show that these functionals are indeed perimeters in an generalised sense and we establish existence of minimisers for the corresponding Plateau's problem; also, when K is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe "flat" boundary conditions.A Γ-convergence result is discussed in the second part of the work. We study the limiting behaviour of the nonlocal perimeters associated with certain rescalings of a given kernel that has faster-than-L 1 decay at infinity and we show that the Γ-limit is the classical perimeter, up to a multiplicative constant that we compute explicitly.
We consider a nonlocal functional J K that may be regarded as a nonlocal version of the total variation. More precisely, for any measurable function u : R d → R, we define J K (u) as the integral of weighted differences of u. The weight is encoded by a positive kernel K, possibly singular in the origin. We study the minimisation of this energy under prescribed boundary conditions, and we introduce a notion of calibration suited for this nonlocal problem. Our first result shows that the existence of a calibration is a sufficient condition for a function to be a minimiser. As an application of this criterion, we prove that halfspaces are the unique minimisers of J K in a ball, provided they are admissible competitors. Finally, we outline how to exploit the optimality of hyperplanes to recover a Γ-convergence result concerning the scaling limit of J K .
We consider nonlocal curvature functionals associated with positive interaction kernels, and we show that local anisotropic mean curvature functionals can be retrieved in a blow-up limit from them. As a consequence, we prove that the viscosity solutions to the rescaled nonlocal geometric flows locally uniformly converge to the viscosity solution to the anisotropic mean curvature motion. The result is achieved by combining a compactness argument and a set-theoretic approach related to the theory of De Giorgi's barriers for evolution equations.
We derive, by means of variational techniques, a limiting description for a class of integral functionals under linear differential constraints. The functionals are designed to encode the energy of a high-contrast composite, that is, a heterogeneous material which, at a microscopic level, consists of a periodically perforated matrix whose cavities are occupied by a filling with very different physical properties. Our main result provides a Γ-convergence analysis as the periodicity tends to zero, and shows that the variational limit of the functionals at stake is the sum of two contributions, one resulting from the energy stored in the matrix and the other from the energy stored in the inclusions. As a consequence of the underlying high-contrast structure, the study is faced with a lack of coercivity with respect to the standard topologies in L p {L^{p}} , which we tackle by means of two-scale convergence techniques. In order to handle the differential constraints, instead, we establish new results about the existence of potentials and of constraint-preserving extension operators for linear, k-th order, homogeneous differential operators with constant coefficients and constant rank.
In the context of image processing, we study a class of integral regularizers defined in terms of spatially inhomogeneous integrands that depend on general linear differential operators. Particularly, the spatial dependence is assumed to be only measurable. The setting is made rigorous by means of the theory of Radon measures and of suitable function spaces modeled on BV. We prove the lower semicontinuity of the functionals at stake and existence of minimizers for the corresponding variational problems. Then, we embed the latter into a bilevel scheme in order to automatically compute the regularization parameters. These parameters are considered to be spatially varying, thus allowing for good flexibility and preservation of details in the reconstructed image. After identifying a series of spatially inhomogeneous regularization functionals commonly used in image processing that are included in our framework, we substantiate its feasibility by performing numerical denoising examples in which the spatial dependence of the integrand is measurable. Specifically, we use Huber versions of the first and second order total variation (and their sum) with both the Huber and the regularization parameter being spatially varying. Notably, the spatially varying version of second order total variation produces high quality reconstructions when compared to regularizations of similar type, and the introduction of the low regularity spatially dependent Huber parameter leads to a further enhancement of the image details. We expect that our theoretical investigations and our numerical feasibility study will support future work on setting up schemes where general differential operators with spatially dependent coefficients will also be part of the optimization scheme.
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