Abstract
We extend classical results on variational inequalities with convex sets with gradient constraint to a new class of fractional partial differential equations in a bounded domain with constraint on the distributional Riesz fractional gradient, the \(\sigma \)-gradient (\(0<\sigma <1\)). We establish continuous dependence results with respect to the data, including the threshold of the fractional \(\sigma \)-gradient. Using these properties we give new results on the existence to a class of quasi-variational variational inequalities with fractional gradient constraint via compactness and via contraction arguments. Using the approximation of the solutions with a family of quasilinear penalisation problems we show the existence of generalised Lagrange multipliers for the \(\sigma \)-gradient constrained problem, extending previous results for the classical gradient case, i.e., with \(\sigma =1\).
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25 March 2021
A Correction to this paper has been published: https://doi.org/10.1007/s00245-021-09760-0
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Acknowledgements
The authors would like to thank to the Reviewers for their helpful comments. The research of J. F. Rodrigues was partially done under the framework of the Project PTDC/MAT-PUR/28686/2017 at CMAFcIO/ULisboa and L. Santos was partially supported by the Centre of Mathematics the University of Minho through the Strategic Project PEst UID/MAT/0003/2013.
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Rodrigues, J.F., Santos, L. On Nonlocal Variational and Quasi-Variational Inequalities with Fractional Gradient. Appl Math Optim 80, 835–852 (2019). https://doi.org/10.1007/s00245-019-09610-0
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DOI: https://doi.org/10.1007/s00245-019-09610-0