Maximal regularity for local minimizers of non-autonomous functionals

By P. Hasto, Jihoon Ok. Journal of the European Mathematical Society (Print), 2019

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We establish local $C^{1,\alpha}$-regularity for some $\alpha\in(0,1)$ and $C^{\alpha}$-regularity for any $\alpha\in(0,1)$ of local minimizers of the functional \[ v\ \mapsto\ \int_\Omega \varphi(x,|Dv|)\,dx, \] where $\varphi$ 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 $\varphi$ in terms of a single condition for the map $(x,t)\mapsto \varphi(x,t)$, rather than separately in the $x$- and $t$-directions. Thus we can obtain regularity results for functionals without assuming that the gap $\frac qp$ between the upper and lower bounds is close to $1$. Moreover, for $\varphi(x,t)$ with particular structure, including $p$-, Orlicz-, $p(x)$- 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a822244ff319ac63dfe68a26cc706d4345eb2448

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Citations in the corpus

Growth conditions and regularity for weak solutions to nonlinear elliptic pdes (2020, unknown): “For related recent references we also quote Duzgun-Marcellini-Vespri [39], [40], Rǎdulescu-Zhang [63], [64], Cencelja-Rădulescu-Repovš [19], Papageorgiou-Rădulescu-Repovš-Dušan [62], Ambrosio-Rădulescu [1], Ragusa-Tachikawa [65], Chlebicka [20], Chlebicka-DeFilippis [21], [22], CupiniGiannetti-Giova”
Recent developments in problems with nonstandard growth and nonuniform ellipticity (2021, background): “The approach in [100] also covers the vectorial case, while the one in [131] is at the moment still confined to the scalar one.”
A variational approach to parabolic equations under general and p,q-growth conditions (2020, background): “Among them there are variational integrands fulfilling a standard growth condition from below and above, functionals of non-standard p, q−growth and double phase (as in [4–6,23,24] and [21,29,34,35]), functionals with exponential growth (see [7,44,46]), variable exponents [21,26,33,36,37] and Orlicz”
Regularity for Double Phase Problems at Nearly Linear Growth (2023, methodology): “The techniques needed for Theorem 1.3 are entirely different from those used in the case of classical superlinear double phase functionals in (1.3) [5,18] and for pointwise uniformly elliptic problems with nonstandard growth conditions [43,44].”
On the Regularity of Minima of Non-autonomous Functionals (2019, unknown): “For this we refer to [36–39, 50].”
A new class of double phase variable exponent problems: Existence and uniqueness (2021, background): “Moreover, recent results for nonuniformly elliptic variational problems and nonautonomous functionals can be found in the papers of Beck-Mingione [11, 12], De Filippis-Mingione [23] and Hästö-Ok [43].”
Local Hölder continuity for fractional nonlocal equations with general growth (2021, unknown): “We also refer to [11, 18, 28, 29, 39] and references therein for the regularity results for equation of the G-Laplacian type.”