Partial Regularity of Strong Local Minimizers in the Multi-Dimensional Calculus of Variations

By Jan Kristensen, Alireza H. Taheri. 2003

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Abstract.Let Ω⊂ℝn be a bounded domain and F:𝕄→ℝ a given strongly quasiconvex integrand of class C2 satisfying the growth condition $${{ |F(\xi)| \le c (1 + |\xi|^p)}}$$ for some c>0 and 2≤p<∞. Consider the multiple integral $${{ I[u] = \int_{{\Omega}} \! F(\nabla u) }}$$ where uW1,p(Ω, ℝN). The main result of the paper is the proof that any strong local minimizer of I[·] is of class C1,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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/abdb9ba29be712b83a9d913cedc2e27f7d91601f

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Progress and puzzles in nonlinear elasticity (2010, background): “Partial regularity of energy minimizers (Evans [15], Kristensen & Taheri [24]) Under similar hypotheses, with a slightly strengthened version of quasiconvexity, and assuming f smooth, any global (or W 1,p-local) minimizer is smooth on the complement of a closed set E of n-dimensional Lebesgue measur”
The Regularity of Critical Points of Polyconvex Functionals (2004, result): “This result was extended by Kristensen & Taheri [ KT01 ] to the case of strong local minimisers (local with respect to variations in W 1,p with p< ∞). Kristensen and Taheri also show that the coun-”
Regularity of minima: An invitation to the dark side of the calculus of variations (2006, background): “More recently, the partial regularity of strong local minimizers of quasiconvex functionals has been proved by Kristensen & Taheri [209].”