Maximal regularity for local minimizers of non-autonomous functionals.(English)Zbl 1485.49044
Summary: 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 \(q/p\) between the upper and lower growth 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.
MSC:
| 49N60 | Regularity of solutions in optimal control |
| 35A15 | Variational methods applied to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35J62 | Quasilinear elliptic equations |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
maximal regularity;non-autonomous functional;variable exponent;double phase;non-standard growth;minimizer;Hölder continuity;generalized Orlicz space;Musielak-Orlicz spaceCite
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