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Remarks about gradient Young measures generated by sequences in Sobolev spaces. (English) Zbl 0828.46031

Brezis, H. (ed.) et al., Nonlinear partial differential equations and their applications. Collège de France Seminar, volume XI. Lectures presented at the weekly seminar on applied mathematics, Paris, France, 1989-1991. Harlow: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 299, 131-150 (1994).
The paper concerns the problems of possible oscillations of solutions of partial differential equations. The oscillatory properties of a sequence of weak convergent functions is described by the parametrized measure or Young measure. It means that there exists a sequence \(f^k\in L^\infty (\Omega; \mathbb{R}^N)\), where \(f^k\to f\) in \(L^\infty (\Omega; \mathbb{R}^N)\) weak* and a family of probability measures \(\nu= (\nu_x )_{x\in \Omega}\) with \(\text{supp } \nu_x \subset \mathbb{R}^N\) such that for every function \(\psi\) continuous in \(\lambda\) and measurable in \(x\), \[ \psi(f^k, x)\to \overline {\psi} (x)= \int_{\mathbb{R}^N} \psi(\lambda, x) d\nu_x (\lambda)\;\text{ in } L^\infty (\Omega)\;\text{ weak}^*, \] where \(\Omega \subset \mathbb{R}^N\) is bounded.
The paper deals with the problems where \(f^k= \nabla u^k\) and \(f^k\) is bounded not in \(L^\infty\) but in \(L^p\).
For the entire collection see [Zbl 0785.00024].

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
28B15 Set functions, measures and integrals with values in ordered spaces
46G10 Vector-valued measures and integration
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