66 problems

Let $\beta\in(0,\pi)$ be the opening angle of the corner, let $\mu$ be the parameter, and let $E_{\mathrm{corner}}(\mu)$ and $E_{\mathrm{corr}}(\mu)$ denote respectively the corner…

Necessity conjecture. This condition is not only sufficient but also necessary: the Flamant solution should not apply for models with quadratic growth in other asymptotic regimes.

Subtransversality conjecture. For alternating projections applied to inconsistent feasibility, subtransversality is necessary for R-linear convergence of the iterates to fixed poin…

Tilt-stability conjecture. Under appropriate assumptions, tilt stability of a local minimizer should exclude the existence of critical multipliers associated with that minimizer.

Let $J_2(p)$ denote the variational problem considered in the source, with exponent $p$. The authors of the cited work conjecture that the nonexistence conjecture for $J_2(p)$.…

Sharpness conjecture. The sharpest condition should instead be

Variational-representation extension conjecture. Similar representations should be derivable for arbitrary cost matrices, still under the finitely discrete setting, using similar t…

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

Let $f_0:\overline{\mathbb{Y}}\to\overline{\mathbb{X}}$ be a finite-distortion homeomorphism with $\mathscr{E}_p[f_0]<\infty$, and let…

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

Let $\mathbb{X}$ and $\mathbb{Y}$ be bounded simply connected Lipschitz domains, let $f_0:\partial\mathbb{Y}\to\partial\mathbb{X}$ be prescribed boundary data, and let … where…

Let $\varepsilon(m)$ be the stability separation curve, and let $m_2^c$ and $m_*$ be as above. Decreasing-branch conjecture. The stability separation curve $\varepsilon(m)$ is decr…

Let $\varepsilon(m)$ be the stability separation curve, let $m_2^c$ be the maximum point of the curve $\varepsilon_2$, and let $m_*$ be the supremum of the positive roots of the cu…

Trade-off invariance principle. For all but some exceptional $\alpha\in[0,+\infty)^m$, if $\inf_{u\in U}H_{\alpha}(u)>-\infty$, then there exist…

Topology conjecture. Locally minimal $k$-partitions $P_1$ and $P_2$ with

Burke–Chen conjecture. The claim that

Let $F$ be the mapping in the generalized nonlinear program, let $g$ be the associated function, and let $Y(\bar x)$ denote the multiplier set at $\bar x$. The notation…

Let ${\bf X}$ be the underlying space, let $f:{\bf X}\to \overline{{\rm I\!R}}$ be a proper function that is prox-regular and subdifferentially continuous, and let $\bar{x}$ be a l…

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…

Consider the heat content functional for an $N$-gon under a fixed-area constraint, with $t\in\mathbb R$, and let the regular $N$-gon be the symmetric critical configuration. Local…

Fix $Q=12$ and consider the Hessian of … where $\ell_Q$ is chosen so that the regular $N$-gon is a critical point. Restrict to eigenvectors orthogonal to the gradient of the area;…

Let $\mathcal{E}_V$ be the Maxwell–Schrödinger energy functional for a spin-$\frac12$ electron, and let $g>0$ be the coupling constant. Two minimizers are identified when they diff…

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…

Let $(\psi,a)$ be the solution in (1.7) constructed under the assumptions of Theorem 1.1, let $h_r$ be the relevant Hermitian metric, and let $b_r$ and $\kappa$ be the parameters a…

Nonexistence conjecture. In general, there exists no optimizer for the infinite-dimensional Weinstein problem.