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\(L^ p\)-approximation of Jacobians. (English) Zbl 0753.46024
The paper investigates the nonlinear function spaces introduced by Giaquinta, Modica and Souček. A vector function \(u=(u_ 1(x),\dots,u_ m(x))\) is said to belong to the nonlinear space \(A^ p(\Omega,R^ m)\) if \[ \| u\|_{A^ p}=\| u\|_{H^{1,p_ 1}(\Omega)}+\sum_{k=2}^ N(\int_ \Omega| M_ k Du(x)|^{p_ k}dx)^{1/p_ k}<\infty, \] where \(\Omega\subset R^ N\), \(M_ k Du(x)\) is the multivector of all minors \[ M_ \alpha^ \beta Du(x)=\text{det}(\partial u^{\beta_ j}(x)/\partial x_{\alpha_ i})_{i,j=1,\dots,k}, \] \(k\leq N\); \(\alpha\), \(\beta\) are multi-indexes from \(I_ k=\{1,\dots,N\}^ k\) and \(J_ k=\{1,\dots,m\}^ k\), \(p=(p_ 1,\dots,p_ N)\), \(1<p_ i<\infty\), \(i=1,\dots,N\). The space \(\text{Cart}^ p(\Omega)\) is defined to be the smallest set in \(A^ p\) which contains \(A^ p\cap C^ 1\) and is closed under weak convergence in \(A^ p\). Here \(u_ n\to u\) weakly in \(A^ p\) if \(u_ n\to u\) weakly in \(H^{1,p_ 1}\) and for each \(k=1,\dots,N\), \(\alpha\in I_ k\), \(\beta\in J_ k\), \(M_ \alpha^ \beta Du_ n\to M_ \alpha^ \beta Du\) weakly in \(L^{p_ k}\). The main result is the following
Theorem. Let \(\Omega\subset R^ N\) be an open set, \(|\Omega|<\infty\) and \(p>(1,\dots,1)\) be a nonincreasing multiexponent. Let \(u\in\text{Cart}^ p(\Omega)\). Then there exists a sequence \((u_ n)_ n\) of \(C^ 1\) functions from \(A^ p(\Omega,R^ m)\) with the following property: \(u_ n\to u\) strongly in \(L^{q_ 1}(\Omega)\), \(M_ i Du_ n\to M_ i Du\) strongly in \(L^{q_ i}\), \(i=1,\dots,N\); \(1\leq q_ i<p\).

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E40 Spaces of vector- and operator-valued functions
74B20 Nonlinear elasticity
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