On the global regularity for minimizers of variational integrals: splitting-type problems in 2D and extensions to the general anisotropic setting. (English) Zbl 1500.49021
Summary: We mainly discuss superquadratic minimization problems for splitting-type variational integrals on a bounded Lipschitz domain \(\Omega \subset{\mathbb{R}}^2\) and prove higher integrability of the gradient up to the boundary by incorporating an appropriate weight-function measuring the distance of the solution to the boundary data. As a corollary, the local Hölder coefficient with respect to some improved Hölder continuity is quantified in terms of the function \({\text{dist}}(\cdot,\partial \Omega)\).The results are extended to anisotropic problems without splitting structure under natural growth and ellipticity conditions. In both cases we argue with variants of Caccioppoli’s inequality involving small weights.
MSC:
| 49N60 | Regularity of solutions in optimal control |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
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