BV functions and parabolic initial boundary value problems on domains

Given a uniformly elliptic second order operator on a possibly unbounded domain , let ( T ( t )) t ≥0 be the semigroup generated by in L 1 ( Ω ), under homogeneous co-normal boundary conditions on ∂ Ω . We show that the limit as t → 0 of the L 1 -norm of the spatial gradient D x T ( t ) u 0 tends to...

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Bibliographic Details
Published inAnnali di matematica pura ed applicata Vol. 188; no. 2
Main Authors Angiuli, Luciana, Miranda, Michele, Pallara, Diego, Paronetto, Fabio
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer-Verlag 2009
Springer Nature B.V
Subjects
Boundary conditions
Boundary value problems
Domains
Mathematics
Mathematics and Statistics
35K20
Linear parabolic equations
47D06
Semigroup theory
functions
49Q15
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Summary:Given a uniformly elliptic second order operator on a possibly unbounded domain , let ( T ( t )) t ≥0 be the semigroup generated by in L 1 ( Ω ), under homogeneous co-normal boundary conditions on ∂ Ω . We show that the limit as t → 0 of the L 1 -norm of the spatial gradient D x T ( t ) u 0 tends to the total variation of the initial datum u 0 , and in particular is finite if and only if u 0 belongs to BV(Ω) . This result is true also for weighted BV spaces. A further characterization of BV functions in terms of the short-time behaviour of ( T ( t )) t ≥0 is also given.
ISSN:0373-3114
1618-1891
DOI:10.1007/s10231-008-0076-3