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Local boundedness of minimizers of anisotropic functionals. (English) Zbl 0984.49019

Let \(\Omega\) be an open subset of \(\mathbb R^n\) and let \(f: \Omega\times \mathbb R \times \mathbb R^n \to \mathbb R\) be a Carathéodory function. The author is concerned with the regularity of local minimizers of the functional \[ J(u) = \int_\Omega f(x,u,Du) dx \] under general hypotheses on \(f\). It is well-known that such local minimizers are bounded if there are constants \(q>1\) and \(C>0\) such that \(|p|^q \leq f(x,z,p)\leq C|p|^q\) for large \(p\) and all \((x,z)\). In fact, a more general dependence on \((x,z)\) can be allowed, but the power behavior with respect to \(p\) is the critical element in the standard proofs. The reviewer [Commun. Partial Differ. Equations 16, No. 2/3, 311-361 (1991; Zbl 0742.35028)] showed that the function \(|p|^q\) can be replaced by a more general function \(F(|p|)\) and other authors proved similar results with varying assumptions on the function \(F\), but growth conditions which depend anisotropically on \(p\) were only considered in some extremely special cases. On the other hand, it was shown by M. Giaquinta [Manuscr. Math. 59, 245-248 (1987; Zbl 0638.49005)] that local minimizers of functionals with \(f(x,z,p) = \sum_{i=1}^n |p_i|^{q_i}\) may be unbounded if the exponents \(q_i\) spread too much.
This paper gives a sufficient condition on \(f\) for all local minimizers to be bounded. The author assumes that there are nonnegative real-valued functions \(A\), \(B\), \(a\), and \(b\) along with a constant \(c\) so that \[ A(p)-b(x)B(|z|)-a(x) \leq f(x,z,p) \leq c(A(p)+b(x)B(|z|) + a(|x|)) \] for all \((x,z,p)\). In addition, \(A\) is assumed to be an even Young function and \(B\) is increasing on \([0,\infty)\), and both are assumed to satisfy a \(\Delta_2\) condition. The main point is a sufficient condition on \(A\) for the local minimizers to be bounded and this condition is given in terms of the Sobolev conjugate introduced by the author in her study of anisotropic Sobolev inequalities. The exact condition, and the corresponding restriction on \(a\), \(b\), and \(B\), are too complicated to reproduce here.

MSC:

49N60 Regularity of solutions in optimal control
49J45 Methods involving semicontinuity and convergence; relaxation
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References:

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