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)...

Given a compact Kahler manifold, the space $\mathcal H$ of its (relative) potentials is an infinite dimensional Frechet on which Mabuchi and Semmes have introduced natural connection $\nabla$. We study certain Lagrangians $T\mathcal H$, in particular Finsler metrics, that are parallel with respect to connection. show geodesics $\nabla$ paths least action, prove convexity property action. This g...

In this paper, we consider the following three-species predator-prey problem with prey-taxis and competition:$ \begin{equation*} \left\{ \begin{aligned} &amp;u_t = d_1\Delta u + \chi_1 \nabla \cdot (u\nabla v) r_1u(1-u-kv-b_1w),&amp;x\in \Omega, t&gt;0,\\ &amp;v_t d_2\Delta v r_2 v(1-hu-v-b_2w),&amp;x\in &amp;w_t d_3\Delta w -\chi_2 (w\nabla u)- \chi_3 v)\\ &amp;\qquad\, +r_3 (-1+au+av -w^\sigm...