Relaxation of quasiconvex functional in BV(Ω, ℝ^p) for integrands f(x, u,∇;u)

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, ℝ^p) 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.