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Young Women in Harmonic Analysis and PDE

December 2-4, 2016

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Judith Campos Cordero (University of Augsburg)

Regularity up to the boundary and sufficient conditions for strong local minimality

The question of finding suitable conditions to guarantee that a given map minimizes a functional is a fundamental problem in the Calculus of Variations. It was first solved by Weierstrass in the scalar case and, after developments from Hestenes [5], Taheri [9], Zhang [10], Kristensen & Taheri [8], etc., Grabovsky & Mengesha [7] finally solved the problem for the vectorial case. Their result is framed under the natural quasiconvexity assumptions. It establishes that $C^1$-extremals at which the second variation is strictly positive are, in fact, strong local minimizers. This settled affirmatively a conjecture by Ball [1], according to which a set of sufficient conditions should be based on the notion of quasiconvexity. In this work we present a new proof of the seminal result by Grabovsky & Mengesha. Furthermore, we introduce a full regularity result (up to the boundary), which aims at relaxing the a priori regularity assumption on the extremal. This is in deep connection with further recent results regarding partial boundary regularity for strong local minimizers [3].

References:

[1] J. M. Ball. The calculus of variations and materials science. Quart. Appl. Math., 56(4):719-740, 1998. Current and future challenges in the applications of mathematics (Providence, RI, 1997).
[2] J. M. Ball and J. E. Marsden. Quasiconvexity at the boundary, positivity of the second variation and elastic stability. Arch. Rational Mech. Anal. 1984.
[3] J. Campos Cordero. Boundary regularity and sufficient conditions for strong local minimizers. Preprint: https://arxiv.org/abs/1605.01614
[4] J. Campos Cordero and K. Koumatos. Necessary and sufficient conditions for strong local minimizers on non-smooth domains. Preprint: https://arxiv.org/abs/1603.07626
[5] M. R. Hestenes. Sufficient conditions for multiple integral problems in the calculus of variations. Amer. J. Math., 70:239-276, 1948.
[6] L. C. Evans. Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal., 95(3):227-252, 1986.
[7] Y. Grabovsky and T. Mengesha. Sufficient conditions for strong local minimal: the case of $C^1$ extremals. Trans. Amer. Math. Soc., 361(3):1495-1541, 2009.
[8] J. Kristensen and A. Taheri. Partial regularity of strong local minimizers in the multi-dimensional Calculus of Variations. Arch. Ration. Mech. Anal., 170(1):63-89, 2003.
[9] A. Taheri. Sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations. Proc. Roy. Soc. Edin, A 131:155-184, 2001.
[10] K. Zhang. Remarks on quasiconvexity and stability of equilibria for variational integrals. Proc. Amer. Math. Soc. 1992.

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