Abstract
We establish interior gradient bounds for functions \({u \in W^1_{1, {\rm loc}} (\Omega)}\) which locally minimize the variational integral \({J [u, \Omega] = \int_\Omega h \left( |\nabla u| \right) dx}\) under the side condition \({u \ge \Psi}\) a.e. on Ω with obstacle \({\Psi}\) being locally Lipschitz. Here h denotes a rather general N-function allowing (p, q)-ellipticity with arbitrary exponents 1 < p ≤ q < ∞. Our arguments are based on ideas developed in Bildhauer et al. (Z Anal Anw 20:959–985, 2001) combined with techniques originating in Fuchs (2011).
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Bildhauer, M., Fuchs, M. Lipschitz regularity for constrained local minimizers of convex variational integrals with a wide range of anisotropy. manuscripta math. 141, 63–83 (2013). https://doi.org/10.1007/s00229-012-0560-3
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DOI: https://doi.org/10.1007/s00229-012-0560-3