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.

在本文中，我们解决了通过用Riesz分数梯度替代经典梯度而获得的非局部矢量变分原理。我们在能量密度的多凸性的主要假设下，证明了贝塞尔分数空间中极小值的存在，并因此证明了非线性分数阶PDE的相关Euler-Lagrange系统解的存在。主要成分是分数Piola同一性，它确定分数梯度的辅因子矩阵的分数散度消失。这种同一性意味着分数梯度行列式的收敛性较弱，进而存在非局部能量极小值的存在。与非线性弹性的局部问题相反，