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Analysis of concentration and oscillation effects generated by gradients. (English) Zbl 0920.49009

The authors study sequences of gradients \((\nabla u_j)_j\) which are bounded in \(L^p(\Omega;M^{m\times N})\), with \(p>1\). Here \(\Omega\) is an open bounded subset of \(R^N\), and \(M^{m\times N}\) is the set of \(m\times N\) matrices. If a function \(\varphi\) on \(R^d\), with \(d=m\times N\), has a growth strictly less than \(p\), the behaviour of \(\varphi(\nabla u_j)\) may be described by the Young-measure \(\nu\) associated with \((\nabla u_j)_j\). If a function \(\psi\) on \(R^d\) grows as the \(p\)-th power, then concentration effects on the limit of \(\psi(\nabla u_j)\) may be described by a measure \(\Lambda\) on \(\Omega\times S^{d-1}\) (\(S^{d-1}\) is the unit sphere in \(R^d\)). The pair \((\nu,\Lambda)\) is called a \(W^{1,p}(\Omega)\)-Young measure-varifold pair associated with the sequence \(( u_j)_j\subset W^{1,p}(\Omega;R^{m})\). The notion of varifold has been introduced by I. Fonseca [ Proc. R. Soc. Edinb., Sect. A 120, No. 1/2, 99–115 (1992; Zbl 0757.49013)]. The authors characterize Young measure-varifold pairs associated with bounded sequences in \(W^{1,p}(\Omega;R^{m})\).

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Citations:

Zbl 0757.49013
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