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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 $$L^{q_ i}$$, $$i=1,\dots,N$$; $$1\leq q_ i<p$$.

##### MSC:
 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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