59 problems

One-dimensionality conjecture. The optimal transition profile will be one-dimensional when $\beta\in(0,1)$ and $|a^+-a^-|$ is small.

Consider the singularly perturbed boundary-value problem for $A(z)$ … with $A(z)\to 0$ as $\operatorname{Re}[z]\to\pm\infty$ along $\operatorname{Im}[z]=0$. At each order, the asym…

Fernández-Real–Ros-Oton conjecture. Suppose that $u:\mathbb{R}^n\to(0,\infty)$ minimizes $I_1$ locally. If $n\leq n_e^*-1$, then $u$ is one-dimensional: in a suitable Euclidean coo…

Aviles–Giga minimizer convergence conjecture. At least when $\Omega$ is convex,

The discussion concerns a family of linear ordinary differential equations exhibiting a singular perturbation, meaning that for certain parameter values the leading coefficient van…

Let $v=v(x,t)$ be a smooth solution of the unperturbed equation … defined for all $x\in\boldsymbol{R}$ and $0\leq t<t_0$, and monotone in $x$ for every $t$. Let $H_f$ be the Hamilt…

Local symmetry conjecture. For any points $P_1,P_2\in C_{\bar n}$, there exist open neighborhoods $\Omega_1,\Omega_2$ and an isometry

Local isometry conjecture. Any small neighborhood $V_1$ of $P_1$ is conjugated to a neighborhood $V_2$ of $P_2$ by an isometry.

Uniqueness and global-maximum conjecture. The coefficients $c_k$ are unique, and whenever the degeneracy cannot be removed, the charged points are charged by a global maximum seque…

Slow-vector-field conjecture. For the delay equation in question, and more generally for families of delay equations in which the delay is treated as a small parameter, the slow ve…

Let $u_\alpha$ be a ground state solution of problem, and let $V_\alpha$ be the associated potential. The maximum of $V_\alpha$ is attained at $1/2$. Concentration conjecture. Grou…

Let $\lambda_\varepsilon”(0)$ denote the numerically computed second derivative of the critical spectral curve, and let $-k\varepsilon^{2/3}$ be its leading-order analytical approx…

Puncture-zone dominance conjecture. A puncture $p_j$ contributes negligibly to Cardy-like asymptotics of the index, except through its near zone and the Gaussian zone.

Nonexistence conjecture. For sufficiently small $\varepsilon_n$, there is no solution of the blow-up equation; equivalently, no Type II spikes exist in this case. This conjecture c…

Let the assumptions of Theorem hold. For $\epsilon>0$, let $(\bm{X}^{(\epsilon)}(t),\bm{w}^{(\epsilon)}(t))$ solve, and let $(\bm{X}^{(0)}(t))$ solve the limiting system with the s…

The linear reaction-convection-diffusion problem is posed on the one-dimensional domain $(0,1)$ with convection coefficient $b(x)=1$, reaction coefficient $c(x)=0$, source term…

Peak-configuration conjecture. The possible configuration of peaks is

The slow-manifold foliation conjecture. For $0<\varepsilon\ll1$, the level sets of $\phi_{{\rm i}\omega}^\varepsilon$ coincide with the leaves of the foliation of the attracting sl…

The transverse-derivative conjecture. As $\varepsilon$ becomes small, $|\partial \angle \phi_{{\rm i}\omega}^\varepsilon/\partial x|$ becomes smaller, while…

Let $f(y;\chi)=0$ denote the polynomial system in the Lorenz nonlinear model, with $y=(y_1,\dots,y_J)$ and parameter $\chi$. The constant solution is $y_j=\chi$ for $j=1,\dots,J$.…

Consider the stochastic version of the Brusselator with the noisy perturbation formulated in the cited stochastic model. In the deterministic Brusselator with large bifurcation par…

Consider a system of particles with arbitrary boundary conditions prescribed on the surfaces of the particles, and let the interaction energy denote the energy beyond the individua…

Let a tropical dynamical system have a tropical singularity at $(u_0,v_0)$ of sink, source, or saddle type. Let $X$ be a neighborhood of $(u_0,v_0)$. Tropical singularity persisten…

Let a tropical dynamical system have a tropical edge $\mathcal E$ that is either of transversal Filippov-sliding or transversal nullcline-sliding type. Let…

Stable critical-point conjecture. Suppose that $u:\mathbb{R}^n\to(0,\infty)$ is a stable critical point of $I_1$ in $\Omega=\mathbb{R}^n$. Then, for $n\leq4$, $u$ is one-dimensiona…