70 problems

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

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 $d\geq 1$, let $U\in L^\infty(\mathbb{T}^d)$, and let $p(t,x,y)$ be the heat kernel associated with the periodic diffusion described by the Dirichlet form in the source. Let…

Assume that $L\mathbf{1}=0$ and $\Lambda_0=0$. Let … be the Taylor expansion of $\Lambda$, and let $M$ be the operator arising in the formal construction, with coefficients $m_\nu$…

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 $Y^\lambda$ be the biased diffusion, let $R^\omega_\lambda(t)$ be defined by … and define its exponentially weighted time average by … For the corrector amplitude … where…

Let $\Sigma_{D,1}$ denote the set of pathological points corresponding to the spectral behavior of $L_0$ at infinity, and let $\operatorname{Res} L_{\varepsilon}$ be the set of sca…

Homogenization conjecture. There exists $\overline{V}\in \operatorname{Lip}_{\mathrm{loc}}(\mathbb{R}^n\times[0,\infty))$ such that $V^\varepsilon$ converges to $\overline{V}$ in t…

Let $u_{0}$ be the solution of the homogenized equation, and let $u_{\mathcal{D}^{\text{p}}}^{\epsilon}$ be the solution of the partially dilation-dilated equation. Motivated by th…

Geometric-scale separation problem. Prove anomalous diffusion for a variant of the construction with geometrically separated active scales.

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…

Let $a_{ij}(\cdot)$ be the coefficient matrix in the framework above, let $\overline{a}_{ij}$ denote its homogenized coefficients, and let $\beta_0$ be the critical value. Write…

Consider sequences of parabolic nonlocal operators of the form … In the local setting, when the sequence of matrix-valued functions is independent of time, the parabolic $H$-limit…

A sequence of monotone operators in fractional divergence form is considered, with the associated fractional fluxes and gradients understood in the setting developed in the paper.…

Let $N$ denote the number of rigid particles and let $\operatorname{vol}[N]$ be their total volume. Let $A=A(p,q)$ be the exponential factor appearing in Theorem 1, arising from th…

Nonstationary homogenization conjecture. According to the scaling regime, the solutions have the following limits: (i) if $\lim_{\epsilon\searrow0}\sigma_\epsilon=+\infty$, they co…

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…

The periodic tubular structure is described by a magnetic Laplacian whose homogenised dispersion relation depends on the perturbation parameter, and for larger positive values of t…

Consider the macroscopic wave-equation approximation for a lattice with i.i.d. random masses, and denote by a.e.d. and a.e.v. the absolute error measures used in the source. Three-…

Let $A$ denote the flow intensity, let $d>0$ be the Markstein number, and let $p=(p_1,p_2)$ be a direction. Write $…$ for the homogenized effective burning velocity in the curvat…

Consider the heterogeneous wave equation … A slow manifold for the corresponding dissipative homogenization problem is not attractive for this wave equation; the source instead con…

Let $a$ denote the heterogeneity amplitude, and let $k_*$ denote the first real convergence-limiting wavenumber singularity of the homogenization series. Second-singularity conject…

Backward-theory homogenization conjecture. A backward-theory approach to the slowly varying theory should support a cognate generalized homogeneous wave equation