Maximal regularity for local minimizers of non-autonomous functionals

: Hästö Peter, Ok Jihoon

Publisher: EUROPEAN MATHEMATICAL SOC-EMS

: 2022

: Journal of the European Mathematical Society

: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY

: J EUR MATH SOC

: 24

: 4

: 1285

: 1334

: 50

: 1435-9855

DOI: https://doi.org/10.4171/JEMS/1118

: https://research.utu.fi/converis/portal/detail/Publication/175031943

: https://arxiv.org/abs/1902.00261v1

We establish local C-1;alpha-regularity for some alpha is an element of (0, 1) and C-alpha-regularity for any alpha is an element of (0, 1) of local minimizers of the functionalnu bar right arrow integral(Omega)phi(x, vertical bar D nu vertical bar)dx,where phi satisfies a(p, q)-growth condition. Establishing such a regularity theory with sharp, general conditions has been an open problem since the 1980s. In contrast to previous results, we formulate the continuity requirement on phi in terms of a single condition for the map(x, t) bar right arrow phi(x, t), rather than separately in the x- and t -directions. Thus we can obtain regularity results for functionals without assuming that the gap q=p between the upper and lower growth bounds is close to 1. Moreover, for phi(x, t) with particular structure, including p-, Orlicz-, p(x)- and double phasegrowth, our single condition implies known, essentially optimal, regularity conditions. Hence, we handle regularity theory for the above functional in a universal way.

1902.00261.pdf