## 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)].

### 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

### Citations:

Zbl 0516.49003; Zbl 0167.107
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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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