On convergence of gradient-dependent integrands |
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Authors: | Martin Kru?ík |
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Institution: | (1) Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 4, CZ-182 08 Praha 8, Czech Republic;(2) Faculty of Civil Engineering, Czech Technical University, Thákurova 7, CZ-166 29 Praha 6, Czech Republic |
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Abstract: | We study convergence properties of {υ(∇u
k
)}k∈ℕ if υ ∈ C(ℝ
m×m
), |υ(s)| ⩽ C(1+|s|
p
), 1 < p < + ∞, has a finite quasiconvex envelope, u
k
→ u weakly in W
1,p
(Ω; ℝ
m
) and for some g ∈ C(Ω) it holds that ∫Ω
g(x)υ(∇u
k
(x))dx → ∫Ω
g(x)Qυ(∇u(x))dx as k → ∞. In particular, we give necessary and sufficient conditions for L
1-weak convergence of {det ∇u
k
}
k∈ℕ to det ∇u if m = n = p.
Dedicated to Jiří V. Outrata on the occasion of his 60th birthday
This work was supported by the grants IAA 1075402 (GA AV ČR) and VZ6840770021 (MŠMT ČR). |
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Keywords: | bounded sequences of gradients concentrations oscillations quasiconvexity weak convergence |
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