Preprint

Inserted: 2 sep 2025
Last Updated: 2 oct 2025

Year: 2025

Abstract:

We prove a stochastic homogenisation result for strongly anisotropic, degenerate integral functionals under suitable moment conditions. Specifically, we study random vectorial functionals whose integrands exhibit degenerate growth and coercivity of order $p>1$ governed by two nonnegative, stationary weight functions $\Lambda$ (governing growth) and $\lambda$ (governing coercivity). We allow the ratio $\Lambda/\lambda$ to become unbounded with positive probability, thereby encompassing the case of strongly anisotropic integrands. Our main result shows that when the integrand is convex in the gradient variable and the weights satisfy the moment condition \[ \mathbb E[\Lambda^\alpha + \lambda^{-\beta}]<+\infty, \] for exponents $\alpha\geq1$, $\beta \geq 1/(p-1)$ such that \[ \frac{1}{\alpha}+\frac{1}{\beta}<\frac{p}{d-1}, \] if $d\geq 3$ (where $d$ is the space-dimension), the functionals almost surely homogenise to a non-degenerate limit energy. Furthermore, in the general non-(quasi)convex case, we prove an analogous homogenisation result under the stricter condition \[ \frac{1}{\alpha}+\frac{1}{\beta}<\frac{1}{d-1}, \] which is shown to be optimal for suitable choices of $p$ and $d$.

Keywords: $\Gamma$-convergence, degenerate growth, stochastic homogenisation, strongly anisotropic materials

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• RegZep25_submitted.pdf