Integral convexity and parabolic systems

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

Publication Type: Journal article

Publication year: 2020


Book Volume: 52

Pages Range: 1489-1525

Journal Issue: 2

DOI: 10.1137/19M1287870


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.

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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.


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

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