Lower semicontinuity and relaxation of signed functionals with linear growth in the context of $${\mathcal A}$$-quasiconvexity
Lower semicontinuity and relaxation of signed functionals with linear growth in the context of $${\mathcal A}$$-quasiconvexity
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$${mathcal A}$$-拟凸线性增长的符号泛函的下半连续性和松弛
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
P. M. Santos
中科院分区：
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作者：
Margarida Baía;Milena Chermisi;José Matias;P. M. Santos
对于 $$\mu \in \mathcal{M}(\Omega; \mathbb{R}^d) \to \int \limits_\Omega f(\mu^a(x))\,{\rm {d}}x +\int \limits_\Omega f^\infty \left( \frac{{\rm{d}}\mu^s}{d|\mu^s|}(x)\right) \, d| \mu^s|(x),$$其中允许序列 {μn} 使得 $${\{{\mathcal{A}}\mu_{n}\}}$$ 在 $${W^{-1 q}_{\rm loc}(\Omega)}$$ 中强烈收敛到零，并且 $${\mathcal {A}}$$ 是具有恒定秩的偏微分算子。被积函数 f 具有线性增长，并且不假设下方的 L∞ 边界。
A lower semicontinuity and relaxation result with respect to weak-* convergence of measures is derived for functionals of the form$$\mu \in \mathcal{M}(\Omega; \mathbb{R}^d) \to \int \limits_\Omega f(\mu^a(x))\,{\rm {d}}x +\int \limits_\Omega f^\infty \left( \frac{{\rm{d}}\mu^s}{d|\mu^s|}(x)\right) \, d| \mu^s|(x),$$where admissible sequences {μn} are such that $${\{{\mathcal{A}}\mu_{n}\}}$$ converges to zero strongly in $${W^{-1 q}_{\rm loc}(\Omega)}$$ and $${\mathcal {A}}$$ is a partial differential operator with constant rank. The integrand f has linear growth and L∞-bounds from below are not assumed.