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A note on the theory of SBV functions. (English) Zbl 0877.49001

The paper is devoted to a new proof of the compactness theorem in SBV, originally proved by L. Ambrosio by an integralgeometric argument [Boll. Unione Mat. Ital., VII Ser. B 3, No. 4, 857-881 (1989; Zbl 0767.49001)]. As in L. Ambrosio [Calc. Var. Partial Differ. Equ. 3, No. 1, 127-137 (1995; Zbl 0837.49011)], where a similar idea was used under more restrictive assumptions, the central point of the new proof is the analysis of the distributional derivative Dψ(u), where uBV and ψ is a Lipschitz function. Given an increasing function f:[0,)[0,] satisfying f(t)/t as t0 and an open set Ω n , it turns out that uSBV(Ω) and S u f(u + -u - )d n-1 < if and only if

sup ψX(f) Dψ(u)-ψ ' (u)λ(Ω)<(1)

for some absolutely continuous measure λ. Here X(f) denotes the collection of Lipschitz and continuously differentiable functions ψ such that

ψ(s)-ψ(t)f(s-t)s,t·

Moreover, if (1) holds, then λ has the approximate differential of u as density.

MSC:
49J10Free problems in several independent variables (existence)
49Q20Variational problems in a geometric measure-theoretic setting
49N60Regularity of solutions in calculus of variations