Conference on Calculus of Variations in Lille - 3rd edition - July 4-6 2022

Exploiting convex duality in the calculus of variations

Jul 6, 2022, 3:00 PM

30m

M2 building, Cité Scientifique - Meeting room, 1st floor (Laboratoire Paul Painlevé)

Speaker

Lukas Koch (MPI for MiS, Leipzig)

Description

I will recall the classical theory of convex duality and explain how this can be used to obtain regularity statements in the study of minimisers of the problem
$${\min }_{u\in W^{1,p}(\mathrm{\Omega })}{\int }_{\mathrm{\Omega }}F(x,\mathrm{D}u)\mathrm{d}x.$$ In particular, I will comment on recent results obtained in collaboration with Cristiana de Filippis (Parma) and Jan Kristensen (Oxford) concerning the validity of the Euler–Lagrange equations for extended real-valued integrands $F$ satisfying no upper growth condition as well as concerning integrands $F$ satisfying controlled duality $(p,q)$-growth. The main example of integrands $F$ satisfying controlled duality $(p,q)$-growth are convex polynomials.