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Local boundedness of minimizers of integrals of the calculus of variations. (English) Zbl 0819.49023

This paper is concerned with regularity properties of local minimizers of \[ J(u)= \int_ \Omega F(x, Du(x)) dx, \] where \(F: \Omega\times \mathbb{R}^ n\to \mathbb{R}\) is convex with respect to \(Du\), but does not satisfy a standard growth condition. It is proved that if \(F(x,.)\) satisfies the so-called \(\Delta_ 2\)-condition, then every local minimizer of \(J\) satisfies a weak form of the Euler-Lagrange equation. This result is next used when \(F(x,\xi)= f(| \xi|)\) and \(f\) is an \(N\)-function satisfying the \(\Delta_ 2\)-condition to prove that local minimizers of \(J\) are locally bounded. The case of functions of the form \(F(x, \xi)= f(x,| \xi|)\) with \(mt^ p\leq f(x,t)\leq M(1+ t)^ q\) for every \(t>0\), with \(0<m<M\) and \(1<p<q\), \(p<n\), \(f(x,.)\) satisfies the \(\Delta_ 2\)-condition, is considered in the last section. Under some additional assumptions on \(f\), it is still proved that local minimizers are locally bounded. These results improve previous results obtained by many authors [see, for example, G. Moscariello and L. Nania, Ric. Mat. 40, No. 2, 259-273 (1991; Zbl 0773.49019)].

MSC:

49N60 Regularity of solutions in optimal control
35D10 Regularity of generalized solutions of PDE (MSC2000)

Citations:

Zbl 0773.49019
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References:

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