Abstract
We give a regularity result for local minimizers \({u}:{\Omega \subset {\mathbb{R}}^3 \to {\mathbb{R}}^3}\) of a special class of polyconvex functionals. Under some structure assumptions on the energy density, we prove that local minimizers u are locally bounded. For each component \({u^{\alpha}}\) of u, we first prove a Caccioppoli’s inequality and then apply De Giorgi’s iteration method to get the boundedness of \({u^{\alpha}}\). Our result can be applied to the polyconvex integral
with suitable \({p,q,r > 1}\).
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Cupini, G., Leonetti, F. & Mascolo, E. Local Boundedness for Minimizers of Some Polyconvex Integrals. Arch Rational Mech Anal 224, 269–289 (2017). https://doi.org/10.1007/s00205-017-1074-7
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DOI: https://doi.org/10.1007/s00205-017-1074-7