JournalsjemsVol. 24, No. 4pp. 1285–1334

Maximal regularity for local minimizers of non-autonomous functionals

  • Peter Hästö

    University of Turku, Finland
  • Jihoon Ok

    Sogang University, Seoul, Republic of Korea
Maximal regularity for local minimizers of non-autonomous functionals cover
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Abstract

We establish local C1,αC^{1,\alpha}-regularity for some α(0,1)\alpha\in(0,1) and CαC^{\alpha}-regularity for any α(0,1)\alpha\in\nobreak (0,1) of local minimizers of the functional

v  Ωϕ(x,Dv)dx,v\ \mapsto\ \int_\Omega \phi(x,|Dv|)\,dx,

where ϕ\phi satisfies a (p,q)(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 ϕ\phi in terms of a single condition for the map (x,t)ϕ(x,t)(x,t)\mapsto \phi(x,t), rather than separately in the xx- and tt-directions. Thus we can obtain regularity results for functionals without assuming that the gap q/pq/p between the upper and lower growth bounds is close to 11. Moreover, for ϕ(x,t)\phi(x,t) with particular structure, including pp-, Orlicz-, p(x)p(x)- and double phase-growth, our single condition implies known, essentially optimal, regularity conditions. Hence, we handle regularity theory for the above functional in a universal way.

Cite this article

Peter Hästö, Jihoon Ok, Maximal regularity for local minimizers of non-autonomous functionals. J. Eur. Math. Soc. 24 (2022), no. 4, pp. 1285–1334

DOI 10.4171/JEMS/1118