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In this paper we address nonlocal vector variational principles obtained by substitution of the classical gradient by the Riesz fractional gradient. We show the existence of minimizers in Bessel fractional spaces under the main assumption of polyconvexity of the energy density, and, as a consequence, the existence of solutions to the associated Euler–Lagrange system of nonlinear fractional PDE. The main ingredient is the fractional Piola identity, which establishes that the fractional divergence of the cofactor matrix of the fractional gradient vanishes. This identity implies the weak convergence of the determinant of the fractional gradient, and, in turn, the existence of minimizers of the nonlocal energy. Contrary to local problems in nonlinear elasticity, this existence result is compatible with solutions presenting discontinuities at points and along hypersurfaces.
Keywords: Nonlocal variational problems, Riesz fractional gradient, Fractional Piola identity, Polyconvexity
Bellido, José C. 1 ; Cueto, Javier 1 ; Mora-Corral, Carlos 2
@article{AIHPC_2020__37_4_955_0,
author = {Bellido, Jos\'e C. and Cueto, Javier and Mora-Corral, Carlos},
title = {Fractional {Piola} identity and polyconvexity in fractional spaces},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {955--981},
publisher = {Elsevier},
volume = {37},
number = {4},
year = {2020},
doi = {10.1016/j.anihpc.2020.02.006},
mrnumber = {4104831},
zbl = {1442.35445},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.02.006/}
}
TY - JOUR AU - Bellido, José C. AU - Cueto, Javier AU - Mora-Corral, Carlos TI - Fractional Piola identity and polyconvexity in fractional spaces JO - Annales de l'I.H.P. Analyse non linéaire PY - 2020 SP - 955 EP - 981 VL - 37 IS - 4 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.02.006/ DO - 10.1016/j.anihpc.2020.02.006 LA - en ID - AIHPC_2020__37_4_955_0 ER -
%0 Journal Article %A Bellido, José C. %A Cueto, Javier %A Mora-Corral, Carlos %T Fractional Piola identity and polyconvexity in fractional spaces %J Annales de l'I.H.P. Analyse non linéaire %D 2020 %P 955-981 %V 37 %N 4 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.02.006/ %R 10.1016/j.anihpc.2020.02.006 %G en %F AIHPC_2020__37_4_955_0
Bellido, José C.; Cueto, Javier; Mora-Corral, Carlos. Fractional Piola identity and polyconvexity in fractional spaces. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 4, pp. 955-981. doi: 10.1016/j.anihpc.2020.02.006
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