We consider a class of non-local functionals recently introduced by H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung, which offers a novel way to characterize functions with bounded ...variation.
We give a positive answer to an open question related to these functionals in the case of functions with bounded variation. Specifically, we prove that in this case the liminf of these functionals can be estimated from below by a linear combination in which the three terms that sum up to the total variation (namely the total variation of the absolutely continuous part, of the jump part and of the Cantor part) appear with different coefficients. We prove also that this estimate is optimal in the case where the Cantor part vanishes, and we compute the precise value of the limit in this specific scenario.
In the proof we start by showing the results in dimension one by relying on some measure theoretic arguments in order to identify sufficiently many disjoint rectangles in which the difference quotient can be estimated, and then we extend them to higher dimension by a classical sectioning argument.
In this paper we introduce a new definition of BV based on measure upper gradients and prove the equivalence of this definition, and the coincidence of the corresponding notions of total variation, ...with the definitions based on relaxation of L1 norm of the slope of Lipschitz functions or upper gradients. As in the previous work by the first author with Gigli and Savaré in the Sobolev case, the proof requires neither local compactness nor doubling and Poincaré.
We analyse a solution method for minimization problems over a space of
$ \mathbb {R}^d $
R
d
-valued functions of bounded variation on an interval I. The presented method relies on piecewise constant ...iterates. In each iteration, the algorithm alternates between proposing a new point at which the iterate is allowed to be discontinuous and optimizing the magnitude of its jumps as well as the offset. A sublinear
$ \mathcal {O}(1/k) $
O
(
1
/
k
)
convergence rate for the objective function values is obtained in general settings. Under additional structural assumptions on the dual variable, this can be improved to a locally linear rate of convergence
$ \mathcal {O}(\zeta ^k) $
O
(
ζ
k
)
for some
$ \zeta \lt 1 $
ζ
<
1
. Moreover, in this case, the same rate can be expected for the iterates in
$ L^1(I;\mathbb {R}^d) $
L
1
(
I
;
R
d
)
.
We study a characterization of BV and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincaré inequality. Compared with previous works, ...we consider more general functionals. We also give a counterexample in the case p=1 demonstrating that, unlike in Euclidean spaces, in metric measure spaces the limit of the nonlocal functionals is only comparable, not necessarily equal, to the variation measure ‖Df‖(Ω).
In this paper we study the regularity properties of fractional maximal operators acting on BV-functions. We establish new bounds for the derivative of the fractional maximal function, both in the ...continuous and in the discrete settings.
In this paper we investigate some questions related to the continuity of maximal operators in W1,1 and BV spaces, complementing some well-known boundedness results. Letting M˜ be the one-dimensional ...uncentered Hardy–Littlewood maximal operator, we prove that the map f↦(M˜f)′ is continuous from W1,1(R) to L1(R). In the discrete setting, we prove that M˜:BV(Z)→BV(Z) is also continuous. For the one-dimensional fractional Hardy–Littlewood maximal operator, we prove by means of counterexamples that the corresponding continuity statements do not hold, both in the continuous and discrete settings, and for the centered and uncentered versions.
In this paper we prove several formulae that enable one to capture the singular portion of the measure derivative of a function of bounded variation as a limit of non-local functionals. One special ...case shows that rescalings of the fractional Laplacian of a function u∈SBV converge strictly to the singular portion of Du.
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