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Linearization at infinity and Lipschitz estimates for certain problems in the calculus of variations. (English) Zbl 0602.49029

Given a bounded open set \(\Omega \subset {\mathbb{R}}^ n\) and a \(C^ 2\) function \(F: M^{n\times N}\to {\mathbb{R}}\), where \(M^{n\times N}\) denotes the space of all \(n\times N\) matrices, with \(n>2\), \(N>1\), let us consider the integral functional \[ I(v)=\int_{\Omega}F(Dv)dy,\quad v\in H^ 1(\Omega,{\mathbb{R}}^ n). \] In this note the authors study the regularity of the minimizers of I. They prove that these minimizers are locally Lipschitzian under assumptions concerning only the convexity and the growth of F at infinity. In order to obtain this result, they make use of a ”blow up” argument resulting in a ”linearization at infinity” [cf. M. Giaquinta, ”Multiple integrals in the calculus of variations and nonlinear elliptic systems”, Ann. Math. Stud. 105 (1983; Zbl 0516.49003) and E. Giusti and M. Miranda, Arch. Ration. Mech. Anal. 31, 173-184 (1968; Zbl 0167.107)].
Reviewer: A.Salvadori

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
35D10 Regularity of generalized solutions of PDE (MSC2000)
35J65 Nonlinear boundary value problems for linear elliptic equations
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References:

[1] Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems (1983) · Zbl 0516.49003
[2] DOI: 10.1007/BF00282679 · Zbl 0167.10703
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