Integral convexity and parabolic systems

Bogelein V, Dacorogna B, Duzaar F, Marcellini P, Scheven C (2020)

Publication Type: Journal article

Publication year: 2020

Journal

SIAM Journal on Mathematical Analysis Society for Industrial and Applied Mathematics

Book Volume: 52

Pages Range: 1489-1525

Journal Issue: 2

DOI: 10.1137/19M1287870

Abstract

In this work we give optimal, i.e., necessary and sufficient, conditions for integrals of the calculus of variations to guarantee the existence of solutions-both weak and variational solutions-to the associated L²-gradient flow. The initial values are merely L² functions with possibly infinite energy. In this context, the notion of integral convexity, i.e., the convexity of the variational integral and not of the integrand, plays the crucial role; surprisingly, this type of convexity is weaker than the convexity of the integrand. We demonstrate this by means of certain quasi-convex and nonconvex integrands.

Authors with CRIS profile

Frank Duzaar Lehrstuhl für Partial Differential Equations

Involved external institutions

Universität Salzburg (Paris Lodron Universität Salzburg)

Austria (AT) École Polytechnique Fédérale de Lausanne (EPFL)

Switzerland (CH) Università degli Studi di Firenze / University of Florence

Italy (IT) Universität Duisburg-Essen (UDE)

Germany (DE)

How to cite

APA:

Bogelein, V., Dacorogna, B., Duzaar, F., Marcellini, P., & Scheven, C. (2020). Integral convexity and parabolic systems. SIAM Journal on Mathematical Analysis, 52(2), 1489-1525. https://doi.org/10.1137/19M1287870

MLA:

Bogelein, Verena, et al. "Integral convexity and parabolic systems." SIAM Journal on Mathematical Analysis 52.2 (2020): 1489-1525.

BibTeX: Download