Asymptotic behavior of the W 1 / q , q -norm of mollified B V functions and applications to singular perturbation problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 77

Motivated by results of Figalli and Jerison [J. Funct. Anal. 266 (2014) 1685–1701] and Hernández [Pure Appl. Funct. Anal., Preprint https://arxiv.org/abs/1709.08262 (2017)], we prove the following formula:

$$

where Ω ⊂ ℝ$$ is a regular domain, u ∈ BV (Ω) ∩ L$$(Ω), q > 1 and η$$(z) = ε$$η(z∕ε) is a smooth mollifier. In addition, we apply the above formula to the study of certain singular perturbation problems.

DOI : 10.1051/cocv/2019051
Classification : 46E35
Keywords: Function of bounded variations, mollifier, fractional Sobolev norm, singular perturbation functional 
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     author = {Poliakovsky, Arkady},
     title = {Asymptotic behavior of the $W^{1 / q,q}$\protect\emph{}-norm of mollified $BV$\protect\emph{} functions and applications to singular perturbation problems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     year = {2020},
     publisher = {EDP Sciences},
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Poliakovsky, Arkady. Asymptotic behavior of the $W^{1 / q,q}$-norm of mollified $BV$ functions and applications to singular perturbation problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 77. doi: 10.1051/cocv/2019051

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