14 problems

Sharp zero-constant conjecture. Proposition should remain true with $C=0$; that is,

Let $f_0:\overline{\mathbb{D}}\to\overline{\mathbb{D}}$ be the prescribed boundary homeomorphism, let $1<p<\infty$, and let…

Let $p\in(2,6)$ and let $\nu>0$ be a mass. An energy ground state is a positive solution of the NLS variational problem on the $\mathcal{T}$-metric graph that minimizes the energy…

Let $(\psi\equiv 0,a=a^{b_r})$ be the constant-curvature solution (1.3), let $(\psi,a)$ be the solution (1.7), and let $h_r$ be the relevant Hermitian metric. Let $b_r$ and…

The $n^{-1}$ convergence-rate conjecture. Under these assumptions, the distance between the convergent subsequence of finite-depth minimisers and its limit is $O(n_k^{-1})$. This c…

Let $G$ be the energy function and let $M_2(\lambda,c_0)$ denote the corresponding supremum, with parameters $\lambda>0$ and $c_0>0$. Quadratic-supremum conjecture. For all…

Let $A(1,R)$ and $A(1,R_\ast)$ be annuli, and let $\rho$ be the weight defining $(\rho,n)$-harmonicity. For an increasing radial diffeomorphism between them, write … and set…

Injectivity conjecture. The mapping $\omega\mapsto\beta(\omega)$ is injective; equivalently, ground state solitons are uniquely identified by the ratio $\beta(\omega)$. The claim i…

Let $P_\omega$ denote the ground state soliton for the cubic-quintic nonlinear Schrödinger equation on $\mathbb{R}^3$, and let $M(P_\omega)$ be its mass. Mass-curve conjecture. The…

Let $E(L)$ be the limiting energy defined by … and let $\lambda$ be the function introduced in the source. For $b>0$, define $E^{\rm 1D}(b)=\mathfrak{b}(\alpha_0,b)$, where…

Peakedness conjecture. Within the allowable range of exponents,

Equivalence of metric regularity and strong metric regularity. Under these hypotheses, metric regularity and strong metric regularity of the subdifferential are the same notions.

Let $F$ define a differential inclusion on $[0,T]$ and let $R:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$ be the reachable map, with reference points $(\bar{x},\bar{y})$. For th…

Finite-support minimizer conjecture. For any $\tau\geq 1$, there is a minimizer $\rho\in\mathfrak{M}$ of the variational problem on the sphere which is a weighted counting measure…