Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form $\partial u_{\epsilon }^{\omega }/\partial t+1/{ϵ}_3𝒞\left(T_3(x/{\epsilon }_3){\omega }_3\right)·\nabla u_{\epsilon }^{\omega }-div\left(\alpha \left(T_1(x/{\epsilon }_1){\omega }_1,T_2(x/{\epsilon }_2){\omega }_2,t\right)\nabla u_{\epsilon }^{\omega }\right)=f$. It is shown, under certain structure assumptions on the random vector field $𝒞\left({\omega }_3\right)$ and the random map $\alpha ({\omega }_1,{\omega }_2,t)$, that the sequence $\left\{u_{ϵ}^{\omega }\right\}$ of solutions converges in the sense of G-convergence of parabolic operators to the solution $u$ of the homogenized problem $\partial u/\partial t-div\left(ℬ\right(t\left)\nabla u\right)=f$.

Given an open set $\mathrm{\Omega }$ of ${\mathbb{R}}^N$ $\left(N>2\right)$, bounded or unbounded, and a function $w\in L^{\frac{N}{2}}\left(\mathrm{\Omega }\right)$ with $w^{+}\ne 0$ but allowed to change sign, we give a short proof that the positive principal eigenvalue of the problem $$-\mathrm{△}u=\lambda w\left(x\right)u,u\in {\mathcal{D}}_0^{1,2}\left(\mathrm{\Omega }\right)$$ is unique and simple. We apply this result to study unbounded Palais-Smale sequences as well as to give a priori estimates on the set of critical points of functionals of the type $$I\left(u\right)=\frac{1}{2}{\int }_{\mathrm{\Omega }}{\left|\nabla u\right|}^{2}dx-{\int }_{\mathrm{\Omega }}G\left(x,u\right)dx,u\in {\mathcal{D}}_0^{1,2}\left(\mathrm{\Omega }\right),$$ when $G$ has a subquadratic growth at infinity.

In this paper, we consider shape optimization problems for the principal eigenvalues of second order uniformly elliptic operators in bounded domains of ${ℝ}^n$. We first recall the classical Rayleigh-Faber-Krahn problem, that is the minimization of the principal eigenvalue of the Dirichlet Laplacian in a domain with fixed Lebesgue measure. We then consider the case of the Laplacian with a bounded drift, that is the operator $-\Delta +v·\nabla$, for which the minimization problem is still well posed. Next, we...