We review some recent results in quantitative stochastic homogenization for divergence-form, quasilinear elliptic equations. In particular, we are interested in obtaining $L^{\infty }$-type bounds on the gradient of solutions and thus giving a demonstration of the principle that solutions of equations with random coefficients have much better regularity (with overwhelming probability) than a general equation with non-constant coefficients.

We study a comparison principle and uniqueness of positive solutions for the homogeneous Dirichlet boundary value problem associated to quasi-linear elliptic equations with lower order terms. A model example is given by $-\Delta u+\lambda \frac{{\left|\nabla u\right|}^2}{u^r}=f\left(x\right),\lambda ,r>0.$ The main feature of these equations consists in having a quadratic gradient term in which singularities are allowed. The arguments employed here also work to deal with equations having lack of ellipticity or some dependence on u in the right hand side. Furthermore, they...

The aim of this paper is to present new existence results for $\phi$-Laplacian boundary value problems with linear bounded operator conditions. Existence theorems are obtained using the Schauder and the Krasnosel’skii fixed point theorems. Some examples illustrate the results obtained and applications to multi-point boundary value problems are provided.