• Corpus ID: 244921058

Representation of the total variation as a $\Gamma$-limit of $BMO$-type seminorms

@inproceedings{ArroyoRabasa2021RepresentationOT,
  title={Representation of the total variation as a \$\Gamma\$-limit of \$BMO\$-type seminorms},
  author={Adolfo Arroyo-Rabasa and Paolo Bonicatto and Giacomo Del Nin},
  year={2021}
}
We address a question raised by Ambrosio, Bourgain, Brezis, and Figalli, proving that the Γ-limit, with respect to the Lloc topology, of a family of BMO-type seminorms is given by 1 4 times the total variation seminorm. Our method also yields an alternative proof of previously known lower bounds for the pointwise limit and conveys a compactness result in Lloc in terms of the boundedness of the BMO-type seminorms. MSC (2020): 26B30, 26D10 (primary); 26A45, 49J45 (secondary). 

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References

SHOWING 1-10 OF 19 REFERENCES
BMO‐Type Norms Related to the Perimeter of Sets
In this paper we consider an isotropic variant of the BMO‐type norm recently introduced (Bourgain, Brezis, and Mironescu, 2015). We prove that, when considering characteristic functions of sets, this
A new function space and applications
We define a new function space $B$, which contains in particular BMO, BV and $W^{1/p,p}$, $p\in (1,\infty)$. We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several
Another look at Sobolev spaces
The standard seminorm in the space $W^{s,p}$, with $s$<$1$, does not converge, when $s$ approaches $1$, to the corresponding $W^{1,p}$ seminorm. We prove that continuity is restored provided we
Γ-convergence , Sobolev norms , and BV functions
We prove that the family of functionals (Iδ) defined by Iδ(g) = ∫∫ RN×RN |g(x)−g(y)|>δ δ |x− y|N+p dx dy, ∀ g ∈ L(R ), for p ≥ 1 and δ > 0, Γ-converges in L(R ), as δ goes to 0, when p ≥ 1. Hereafter
Quasi-convex integrands and lower semicontinuity in L 1
In this paper it is shown that, under mild continuity and growth hypotheses, if $f(x,u,.)$ is quasi-convex and if $u_n $, $u \in W^{1,1} $ are such that $u_n \to u$ in $L^1 $, then \[ \int_\Omega
BMO-type seminorms and Sobolev functions
Following some ideas of a recent paper by Bourgain, Brezis and Mironescu, we give a representation formula of the norm of the gradient of a Sobolev function which does not make use of the
On formulae decoupling the total variation of BV functions
FUNCTIONS OF BOUNDED VARIATION AND FREE DISCONTINUITY PROBLEMS (Oxford Mathematical Monographs)
By Luigi Ambrosio, Nicolo Fucso and Diego Pallara: 434 pp., £55.00, isbn 0-19-850254-1 (Clarendon Press, Oxford, 2000).
An Introduction to-convergence
1. The direct method in the calculus of variations.- 2. Minimum problems for integral functionals.- 3. Relaxation.- 4. ?-convergence and K-convergence.- 5. Comparison with pointwise convergence.- 6.
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