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Young Women in Harmonic Analysis and PDE

December 2-4, 2016

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].


[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:
[4] J. Campos Cordero and K. Koumatos. Necessary and sufficient conditions for strong local minimizers on non-smooth domains. Preprint:
[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.


Prof. Daniel Huybrechts erhält gemeinsam mit Debarre, Macri und Voisin ERC Synergy Grant

8. November 2019: "Pentagramma Mirificum". Hirzebruch lecture by Sergey Fomin

Hausdorff-Kolloquium im WS 2019/20

Toeplitz Kolloquium zur "Didaktik und Geschichte der Mathematik" im WS 2019/20

Berufspraktisches Kolloquium im WS 2019/20

21. Oktober 2019: Plücker Lecture 2019 by Frank den Hollander

Prof. Peter Scholze erhält Verdienstorden der Bundesrepublik Deutschland

Prof. Dr. Valentin Blomer wurde zum Mitglied der Academia Europaea gewählt

Mathe-Team der Uni Bonn erzielt Spitzenplatz bei internationalem Wettbewerb

Ausschreibung: W2-Professur Reine Mathematik (Bewerbungsschluss: 31. Juli 2019)

Prof. Jan Schröer erhält Lehrpreis der Fakultät 2018; Sonderpreis für Dr. Antje Kiesel

Prof. Peter Scholze erhält Fields-Medaille 2018

Bonner Mathematik weiterhin exzellent

Prof. Stefan Schwede zum Fellow of the AMS gewählt

Bonner Mathematik im Shanghai-Ranking auf Platz 36 und bundesweit führend

Prof. Catharina Stroppel wurde zum Mitglied der Nationalen Akademie der Wissenschaften Leopoldina gewählt

Prof. Peter Scholze neuer Direktor am MPIM

Dr. Thoralf Räsch erhält Lehrpreis der Uni Bonn

Bonner Mathematik beim CHE-Ranking wieder in Spitzengruppe

Bonner Mathematik beim QS World University Ranking 2018 weltweit unter den TOP 50 platziert und bundesweit führend

Prof. Peter Scholze wurde zum Mitglied der Nationale Akademie der Wissenschaften Leopoldina und der Berlin-Brandenburgische Akademie der Wissenschaften gewählt.

Prof. Peter Scholze erhält den Gottfried Wilhelm Leibniz-Preis 2016

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