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Maximum principle for vector-valued minimizers of some integral functionals. (English) Zbl 0729.49015
In the unit ball $$\Omega \subset {\mathbb{R}}^ n$$ the author considers the functional $F(u)=\int_{\Omega}\{| Du|^ p+a| Du|^ 2+b| \det Du|^ q+c| adj Du|^ q\}dx,$ where u: $$\Omega\to {\mathbb{R}}^ n$$, Du is the $$n\times n$$ matrix of the gradient of u and adj Du is the adjoint matrix (i.e. the matrix of algebraic complements). The constants a, b, c, p and q are nonnegative and $$p\geq 1$$. Let $$u=(u^ 1,...,u^ n)\in W^{1,1}(\Omega)$$ be a minimizer of F (i.e. $$F(u)\leq F(u+\phi)$$ for any $$\phi \in W_ 0^{1,1}(\Omega)$$ and $$F(u)<\infty)$$. It is proved that if $$| u^ i| \leq k$$ on $$\partial \Omega$$ for some $$i\in \{1,...,n\}$$, then $$| u^ i| \leq k$$ on $$\Omega$$.

##### MSC:
 49J40 Variational inequalities 35J50 Variational methods for elliptic systems 35B50 Maximum principles in context of PDEs