ISSN:
1432-0673
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract In this paper it is shown that if p(x, u,·) is a quasiconvex function with linear growth, then the relaxed functional in BV(Ω, ℝp) of $$u \to \int\limits_\Omega {f(x, u(x), \nabla u(x)) dx} $$ with respect to the L 1 topology has an integral representation of the form $$\begin{gathered} \mathfrak{F}(u) = \int\limits_\Omega {f(x, u(x), \nabla u(x)) dx + \int\limits_{\Sigma (u)} {K(x, u^ - (x), u^ + (x), v(x)) dH_{N - 1} (x)} } \hfill \\ + \int\limits_\Omega {f^\infty (x, u(x),dC(u))} \hfill \\ \end{gathered} $$ where Du = ∇u dx + u +−u −)⊗v dH N−1L∑(u)+C(u). The proof relies on a blow-up argument introduced by Fonseca & Müller in the case where u ∈ W 1,1 and on a recent result by Alberti showing that the Cantor part C(u) is rank-one valued.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00386367
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