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

Chipot, Michel; Evans, Lawrence C.

Proc. R. Soc. Edinb., Sect. A 102, 291-303 (1986).

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

Keywords:

integral functional; regularity; minimizers; ”blow up” argument; linearization at infinity

Citations:

Zbl 0516.49003; Zbl 0167.107

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 · doi:10.1007/BF00282679