Relaxation of quasiconvex functional in BV(Ω, ℝp) for integrands f(x, u,∇;u)
Relaxation of quasiconvex functional in BV(Ω, ℝp) for integrands f(x, u,∇;u)
复制标题
BV(Ω, ℝp) 中被积函数 f(x, u,∇;u) 的拟凸泛函的松弛
DOI：
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发表时间：
1993
期刊：
影响因子：
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通讯作者：
S. Müller
中科院分区：
文献类型：
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作者：
I. Fonseca;S. Müller
AbstractIn 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 L1 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 dHN−1L∑(u)+C(u). The proof relies on a blow-up argument introduced by Fonseca & Müller in the case where u ∈ W1,1 and on a recent result by Alberti showing that the Cantor part C(u) is rank-one valued.
AbstractIn 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 L1 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 dHN−1L∑(u)+C(u). The proof relies on a blow-up argument introduced by Fonseca & Müller in the case where u ∈ W1,1 and on a recent result by Alberti showing that the Cantor part C(u) is rank-one valued.