114 problems

Coincidence conjecture. If $\Omega$ is a convex $C^{1,1}$ domain, then

Let $(X,\omega)$ be a compact Kähler manifold, let $(\mathcal E^1,d_1)$ be the metric completion of the space of Kähler potentials, and let the extended Mabuchi K-energy be a map…

For $\beta<0$, let $(u(r),f(r)e^{\sqrt{-1}d\theta})$ be a radially symmetric solution of the two-component nonlinear Schrödinger system described above, with $d\in\mathbb{N}$ and…

Let $\mu_{c,k}$, respectively $N_{c,k}$, denote the $k$-th eigenvalue of the operator $H_c+V$, not counted with multiplicity, respectively its multiplicity. Let $N$ be the number o…

Global mountain-pass minimizer conjecture. The global mountain pass solution is also a global constrained minimizer of the strain energy.

The Sherrington–Kirkpatrick model has a Parisi functional whose argument is a functional order parameter. Panchenko's convexity conjecture. The Parisi functional is convex in the f…

Optimality conjecture. The choice of trial wave function used in the proof is optimal, in the sense that Equation holds as equality apart from errors of order…

Collective-field approximation conjecture. For the more general problems considered in this paper, $F_{\phi}(v)$ is close to $F_\infty(v)$.

Attainment conjecture. Both $\mathcal{S}_{d,2^*,\varepsilon}$ and $\tilde{\mathcal{J}}_{p,\mathcal{G}^d_\varepsilon}$ are attained.

Let $N=2$ and let $m>1$. For each positive mass $\mu>0$, consider the polyharmonic Dirichlet and Navier boundary-value problems under assumptions (FP1)--(FP4), with the normalizati…

Let the curves in Figure be energy curves for the problem, with solutions parametrized by their energy and with parameter $\lambda\in\mathbb{R}$. The conjectured joined diagram is…

Existence conjecture. For every energy level $k\in(U_*,c)$, there exists a contractible or non-contractible closed $(g,\sigma,U)$-geodesic.

Let the energy be studied along the line $\gamma=2\pi\beta\kappa$, with real-valued minimizers and $\kappa=\pm 1$. Linear interpolation conjecture. The energy of the real-valued mi…

Let $\mathcal{E}$ be the energy functional for spherical ferromagnets, let $H_{0,2}$ be the profile class used for the axisymmetric construction, and let $\kappa\geq 4$. Uniqueness…

Let $\mathcal{E}$ be the energy functional for spherical ferromagnets, and let $\kappa_0\geq 4$ be such that for every $\kappa>\kappa_0$ there exists a first-type saddle point of…

Pure-power nonlinearity conjecture. The case $\underline b=0=\overline b$ can occur only when

Consider the convex–concave splitting scheme for the Allen–Cahn equation with either box constraints or an $L^1$ penalty, in the setting where the source establishes the correspond…

Small-mass negativity conjecture. In this range, there exists $c_0=c(N,p,q,b)>0$ such that

Let $N$, $q$, $b$, and $p$ be parameters as in the paper, and let obreak 2q^ and $(N-q)^+$ denote the corresponding critical exponents and positive part. For $c0$, let…

Let $m$ and $\alpha$ be the parameters defining the weighted Sobolev quotient $\mathcal{S}(m,2,\alpha,\alpha,\infty)$, and suppose that this quotient admits a $0$-th nonsingular mi…

Existence conjecture. For every $\Lambda\in(4\pi,+\infty)$, $F(u)$ has at least one critical point subject to this constraint. This would differ substantially from the non-existenc…

Non-attainment conjecture. The solution satisfies

Uniqueness conjecture for maximizers. At least for some special rearrangement classes, there exists $\hat v\in\mathcal M$ such that

Let $g$ be the nonlinearity in the normalized nonlinear Schrödinger equation considered in the paper. In the third case, namely … all mountain pass solutions have the corresponding…

Let $J_{\alpha}(u)$ be the functional associated with a $Q$-curvature-type equation, and let $C(\alpha)$ denote a lower-bound constant for this functional. Gui-Li-Wei-Ye's conjectu…