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Added in proof:
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Nitsche, J.C.C. Variational problems with inequalities as boundary conditions or How to fashion a cheap hat for giacometti's brother. Arch. Rational Mech. Anal. 35, 83–113 (1969). https://doi.org/10.1007/BF00247614
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DOI: https://doi.org/10.1007/BF00247614