55 problems

Berestycki–Caffarelli–Nirenberg conjecture. If $\mathbb{R}^n\setminus\overline{\Omega}$ is connected, then the existence of a bounded solution implies that $\Omega$ is a ball, a ha…

Radial symmetry conjecture. The extremal functions in $(\mathrm{cknrs})$ are radially symmetric if

Cazacu–Flynn–Lam's conjecture. The sharp constant satisfies

Let $n\geq 3$, $-\infty<a<\frac{n-2}{2}$, and $a\leq b<a+1$. Consider the critical Caffarelli–Kohn–Nirenberg equation … where $p=\frac{2n}{n-2+2(b-a)}$, and let … be the Felli–Schn…

Let $q_{\mathrm{ic}}=\delta_{-\zeta}+\delta_{\zeta}$ for $\zeta>0$, set $\mathfrak{m}=1$ and $\xi_1=0$, and consider the variational problem in which $T$ and these data are fixed.…

Let $\Omega$ be a domain, let $A$ be the prescribed parameter, and let $D=D_{\alpha,A}$ be an optimal configuration. Say that $D$ has the same symmetries as $\Omega$ when every sym…

Consider an annulus of thickness $\tau$, and write $\delta=A/|\Omega|$ for the prescribed area fraction. Let $D=D_{\alpha,A}$ be an optimal configuration, and let symmetry breaking…

Let $\Omega_h$ be a dumbbell domain, and let $D=D_{\alpha,A}$ be an optimal configuration. Dumbbell symmetry conjecture. For every $\alpha>0$, there is $\rho_1(\alpha,h)>0$ such th…

Let $\Omega_h$ be a dumbbell domain, let $D=D_{\alpha,A}$ be an optimal configuration, and let $\psi_{\Omega}$ be the function whose maxima determine the relevant localization regi…

Fix $\jmath^*>1$ and use the center bundle equations described above. Higher-symmetry discrete-wave conjecture. Generically, there is a deleted neighbourhood…

Fix $\jmath^*\in\mathbb{N}$ and let $(\varepsilon,\mu)\in\mathbb{R}^2$. Consider the center bundle equations … where $v\in\mathbb{C}$, $\omega_{\mathrm{rot}}\neq0$, the functions…

Symmetry-breaking phase conjecture. For $\alpha=0$, outside the class of measures without infinite contours, there exists an infinite number of extreme infinite-volume phases break…

Pemantle–Steif conjecture. For any Heisenberg model on any graph, SB is equivalent to SB+.

Let $\mu_0\in\mathfrak g$ be nonzero, let $\mu:[0,\delta]\to\mathfrak g$ be a regular deformation of $\mu_0$, meaning that $\mu(t)$ is regular for all $t>0$, and set…

For $1<p<N$, consider the critical quasi-linear Hénon equation … where $N\geq2$, $p_\alpha^*=\frac{p(N+\alpha)}{N-p}$, and $\alpha>0$. Gladiali–Grossi–Neves conjecture. Nonradial s…

Gladiali–Grossi–Neves conjecture. Non-radial positive solutions may exist only when $\alpha=\alpha_k$ for some $k\in\mathbb N$.

Let $\psi:\mathbb{R}^2\to\mathbb{C}$ be a vortex solution of the Ginzburg–Landau equation with degree $n$, and call it radially symmetric when its modulus depends only on the radia…

Let $C$ be a convex set that is not a ball, let $g$ be a generic non-constant function on $\partial C$, and let $\mathcal{QS}(f,g)$ denote the corresponding overdetermined free bou…

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. For $v\in H^1(\Omega)\setminus\{0\}$, define … Let $m_0(\Omega)$ be the largest value of $m_0$ such that the solution…

Let $C(N)$ be the sharp constant in the second order Hydrogen uncertainty principle … For $N\in\{2,3\}$, the paper establishes $C(N)<(N+1)^2/4$. The dimension-four strict-bound con…

Symmetry-breaking conjecture. The only manner in which the symmetry of a global extremal can break is when the symmetric extremal function is linearly unstable; consequently, it ca…

Consider the boundary-value problem … with $\lambda>0$ and $l>0$, and its branch of positive solutions. A symmetry-breaking bifurcation point is a point on this branch at which non…

Let $n\geq 2$, $a<0$, $a<b<a+1$, $a<\frac{n-2}{2}$, and $p=\frac{2n}{n-2+2(b-a)}$. Define … where $\Lambda:=\left(\frac{n-2-2a}{2}\right)^2$, and let … The equation…

Let $1<p<N$, let $a_c=(N-p)/p$, and let $a$ and $b$ satisfy $a<a_c$ and $a e?$. The Caffarelli–Kohn–Nirenberg inequality is considered with … For the general case, the optimal symm…

Consider a neural network system with connectivity matrix $H$, and perturb it by replacing $H$ with $H+\epsilon A$, where $A$ is a random perturbation and $\epsilon$ measures its s…