An introduction to $BV$ functions in Wiener spaces

Abstract In this paper we define jump set and approximate limits for BV functions on Wiener spaces and show that the weak gradient admits a decomposition similar to the finite dimensional case. We also define the SBV class of functions of special bounded variation and give a characterisation of SBV via a chain rule and a closure theorem. We also provide a characterisation of BV functions in terms of the short-time behaviour of the Ornstein-Uhlenbeck semigroup following an approach due to Ledoux.

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BV Functions on Convex Domains in Wiener Spaces

Potential Analysis ◽

10.1007/s11118-015-9462-9 ◽

2015 ◽

Vol 43 (1) ◽

pp. 23-48 ◽

Cited By ~ 2

Author(s):

Alessandra Lunardi ◽

Michele Miranda ◽

Diego Pallara

Keyword(s):

Convex Domains ◽

Bv Functions ◽

Wiener Spaces

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Pairings between bounded divergence-measure vector fields and BV functions

Advances in Calculus of Variations ◽

10.1515/acv-2020-0058 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Graziano Crasta ◽

Virginia De Cicco ◽

Annalisa Malusa

Keyword(s):

Vector Field ◽

Conservation Laws ◽

Bounded Variation ◽

Vector Fields ◽

Hyperbolic Conservation Laws ◽

Divergence Measure ◽

Compensated Compactness ◽

Hyperbolic Conservation ◽

Bv Functions ◽

Bounded Functions

AbstractWe introduce a family of pairings between a bounded divergence-measure vector field and a function u of bounded variation, depending on the choice of the pointwise representative of u. We prove that these pairings inherit from the standard one, introduced in [G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 1983, 293–318], [G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147 1999, 2, 89–118], all the main properties and features (e.g. coarea, Leibniz, and Gauss–Green formulas). We also characterize the pairings making the corresponding functionals semicontinuous with respect to the strict convergence in \mathrm{BV}. We remark that the standard pairing in general does not share this property.

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