Quantitative Nonlinear Homogenization: Control of Oscillations

Quantitative stochastic homogenization of linear elliptic operators is by now well-understood. In this contribution we move forward to the nonlinear setting monotone with p-growth. This work dedicated a quantitative two-scale expansion result. By treating range exponents $$2\le p <\infty $$ in dimensions $$d\le 3$$ , are able consider genuinely equations and systems such as $$-\nabla \cdot A(x)(1+|\nabla u|^{p-2})\nabla u=f$$ (with A random, non-necessarily symmetric) for first time. When going from $$p=2$$ $$p>2$$ main difficulty analyze associated linearized operator, whose coefficients degenerate, unbounded, depend on random input via solution equation. One our achievements control intricate dependence, leading annealed Meyers’ estimates which key optimal result establish (this also new periodic setting).