12 problems

Let $B(0,1)\subset\mathbb{R}^d$ be the unit ball. Let $\lambda\in C^\infty(\overline{B(0,1)})$ satisfy $\lambda>0$ on $\overline{B(0,1)}$, and let…

Let $d\geq 2$, let $\nu_{x,y}(\xi)$ be a Young-measure density analytic in $x$ and $y$, discretize $\xi$ algebraically at $N_\xi\sim\varepsilon^{-1}$, and suppose $R_1$ is polynomi…

Let $p\in(1,\infty)$ and $t_{\max}>0$. For each natural number $n$, let $(\hat{k}_n,\hat{\mu}_n)\in L^\infty(\Omega;\mathbb{R}_+)^2$ solve the (vp-IMD) problem, respectively the (s…

Effective diffusivity conjecture. The sequence $u^N$ converges weakly to the stationary solution $u$ of

The space-time conductance IIP conjecture. An IIP holds for the continuous-time Markov chain on $\mathbb Z^d$ with generator

Let $d\ge2$ and let $p,q>d/2$ satisfy $1/p+1/q<2/d$. Consider symmetric long-range conductance configurations $\{c(x,y)=c(y,x):x,y\in\mathbb Z^d\}$ that are averaging and ergodic i…

Let $d\ge1$, let $\Omega^\star$ and $\Omega_{p,q}$ denote the sets of environments defined in the paper, and let $\mathrm{IIP}$ denote the invariance invariance principle establish…

Let $\Omega$ be the domain and let $u^{\varepsilon}$ and $u$ denote, respectively, the solution of the rapidly oscillating elliptic equation and its homogenized limit, with data…

Let $\lambda_{k,\varepsilon}$ and $\lambda_{k,0}$ denote the $k$-th Steklov eigenvalues, respectively, for the rapidly oscillating boundary weight $\rho_\varepsilon$ and its homoge…

Assume that $H(p)=\varphi(|p|)$, where $\varphi:[0,\infty)\to\mathbb{R}$ satisfies (H7), and let … be the associated sequence, with $M_i=\varphi(s_{2i-1})$ and…

Let $S\subset\Omega$ be an open set with closure contained in $\Omega$. Consider continuously graded composites with inclusions having $C^{1,\alpha}$ boundaries, and let…

Let $\Omega$ be the domain, let $A_{\varepsilon}$ and $B_{\varepsilon}$ be the coefficient fields from the preceding boundary-value problem, and let $B^\sharp$ be the homogenized c…