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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 \(\Omega\).

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