A Boundedness Result for Minimizers of Some Polyconvex Integrals (original) (raw)

Partial Regularity of Strong Local Minimizers in the MultiDimensional Calculus of Variations

Archive for Rational Mechanics and Analysis, 2003

Let Ω⊂ℝn be a bounded domain and F:𝕄→ℝ a given strongly quasiconvex integrand of class C 2 satisfying the growth condition \({{ |F(\xi)| \le c (1 + |\xi|^p)}}\) for some c>0 and 2≤p \({{ I[u] = \int_{{\Omega}} \! F(\nabla u) }}\) where uW 1,p (Ω, ℝN ). The main result of the paper is the proof that any strong local minimizer of I[·] is of class C 1,αloc for any α(0,1) on an open set of full n-dimensional measure. In the case of weak local minimizers we establish the same result under the extra assumption that the oscillations in the gradient of the minimizer are not too large. Without such an assumption weak local minimizers need not be partially regular as we show by a class of examples. We also briefly discuss the question of existence of strong local minimizers for I[·] and connections of our results to extensions of Weierstrass’ sufficiency theorem to the multi-dimensional setting.

Regularity of solutions to higher-order integrals of the calculus of variations

International Journal of Systems Science, 2008

We obtain new regularity conditions for problems of calculus of variations with higher-order derivatives. As a corollary, we get non-occurrence of the Lavrentiev phenomenon. Our main regularity result asserts that autonomous integral functionals with a Lagrangian having coercive partial derivatives with respect to the higher-order derivatives admit only minimizers with essentially bounded derivatives.

Regularity of solutions to second-order integral functionals in variational calculus

arXiv preprint arXiv:0707.2404, 2007

Abstract: We obtain regularity conditions of a new type of problems of the calculus of variations with second-order derivatives. As a corollary, we get non-occurrence of the Lavrentiev phenomenon. Our main result asserts that autonomous integral functionals of the calculus of variations with a Lagrangian having superlinearity partial derivatives with respect to the higher-order derivatives admit only minimizers with essentially bounded derivatives.

On non quasiconvex problems of the calculus of variations

Discrete & Continuous Dynamical Systems - A, 2005

We study existence of minimizers for problems of the type inf Ω f (Du (x)) dx : u = u ξ 0 on ∂Ω where f is non quasiconvex and u ξ 0 is an affine function. Applying some new results on differential inclusions, we get sufficient conditions. We also study necessary conditions. We then consider some examples.

Constrained variational calculus: the second variation (part I)

2010

This paper is a direct continuation of arXiv:0705.2362 . The Hamiltonian aspects of the theory are further developed. Within the framework provided by the first paper, the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves. A necessary and sufficient condition for minimality is proved.