Stochastic homogenization of nonconvex discrete energies with degenerate growth
Abstract
We study the continuum limit of discrete, nonconvex energy functionals defined on crystal lattices in dimensions $d\geq 2$. Since we are interested in energy functionals with random (stationary and ergodic) pair interactions, our problem corresponds to a stochastic homogenization problem. In the nondegenerate case, when the interactions satisfy a uniform $p$growth condition, the homogenization problem is wellunderstood. In this paper, we are interested in a degenerate situation, when the interactions neither satisfy a uniform growth condition from above nor from below. We consider interaction potentials that obey a $p$growth condition with a random growth weight $\lambda$. We show that if $\lambda$ satisfies the moment condition $\mathbb E[\lambda^\alpha+\lambda^{\beta}]<\infty$ for suitable values of $\alpha$ and $\beta$, then the discrete energy $\Gamma$converges to an integral functional with a nondegenerate energy density. In the scalar case it suffices to assume that $\alpha\geq 1$ and $\beta\geq\frac{1}{p1}$ (which is just the condition that ensures the nondegeneracy of the homogenized energy density). In the general, vectorial case, we additionally require that $\alpha>1$ and $\frac{1}{\alpha}+\frac{1}{\beta}\leq \frac{p}{d}$. Recently, there has been considerable effort to understand periodic and stochastic homogenization of elliptic equations and integral functionals with degenerate growth, as well as related questions on the effective behavior of conductance models in degenerate, random environments. The results in the present paper are to our knowledge the first stochastic homogenization results for nonconvex energy functionals with degenerate growth under moment conditions.
 Publication:

arXiv eprints
 Pub Date:
 June 2016
 arXiv:
 arXiv:1606.06533
 Bibcode:
 2016arXiv160606533N
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Functional Analysis;
 Mathematics  Probability;
 35B27;
 35J70;
 60F99
 EPrint:
 paper is accepted for publication in SIAM J. on Mathematical Analysis