28 problems

Positivity conjecture. The Green's function $H_{m,M}$ is positive if and only if $M\in(-m,\lambda_1)$.

Let $g$ satisfy the self-similar boundary value problem referred to as, with the boundary conditions described in the preceding discussion, including $g'(0)<0$. Nonexistence conjec…

Let a boundary-value problem be subject to the estimate referenced in the source as … . The conjecture proposes a general regularity consequence of polynomial resolvent growth, ext…

Unique-solvability conjecture. The boundary value problem is uniquely solvable if and only if $\mathcal{N}^\Omega_\zeta=\emptyset$.

A two-point boundary value problem (BVP) for an ordinary differential operator is called dissipative when its associated operator is dissipative, and Birkhoff-regular when its boun…

Let $Q$ be either of the two isospectral mixed Dirichlet-Neumann problems on a quarter-sphere shown in the source, and let $Q_{a,2}$ be the axisymmetric problem with the Dirichlet…

A bounded planar domain is equipped with a decomposition of its boundary into Dirichlet and Neumann parts, and Dirichlet-Neumann isospectrality means that swapping these boundary c…

Thurston's conjecture. There is a unique hyperbolic metric $g$ on $M$ which induces $h$ on the boundary, and for which the boundary is convex.

Let $n\geq 3$, let $\mathbb{B}^n\subset\mathbb{R}^n$ be the unit ball with boundary sphere $\mathbb{S}^{n-1}$, and let $u\in C^{\infty}(\mathbb{B}^n)$ be positive and satisfy … whe…

Let $(M^n,g)$, with $n\geq 3$, be a compact Riemannian manifold with nonnegative Ricci curvature, and let the second fundamental form satisfy $\Pi\geq 1$ on $\partial M$. Let…

Fefferman–Graham flatness conjecture. There exists a metric representative in $(M_+,\boldsymbol{c})$ that is formally to all orders Poincaré–Einstein.

Let ${\mathcal E}$ be the space of solutions modulo the relevant gauge equivalence, let ${\mathcal B}={\mathcal B}_C$ denote the conformal-mean curvature boundary data space, let…

Anderson's conjecture. For each fixed $\Lambda$, the boundary map $\Pi$ is regular at generic metrics, meaning that its linearization is surjective with split kernel at those metri…

Dirichlet infinite-linking convergence conjecture. The constants and minimisers satisfy

Perturbation-limit conjecture. Up to addition of a constant, the smooth solution $u$ is the suitable limit of the sequence of solutions $w_\varepsilon$ as $\varepsilon\to0$.

Let $(M,\bar g,\bar u)$ be an asymptotically flat, static vacuum triple, and let the geometric boundary value problem be … where the interior equations are…

Let $u$ solve the hydrodynamic equation … with boundary parameters determined by the reservoir rates $(\alpha,\beta,\gamma,\delta)(t)$ through the relations defining $\nu_\pm(t)$.…

Regularity conjecture. If $h\in C[0,1]$ or $h\in L^1(0,1)$, then the solution $u$ belongs to $C^1_{2-\alpha}[0,1]$. Moreover, if $h\in C(0,1]$ satisfies $(H)$, then the solution…

Let $a\in L^{\infty}(0,T)$ satisfy the assumptions in (hp-a), and let $\gamma\in\mathbb{R}\setminus\{-1,0,1\}$. Consider the Neumann and periodic boundary value problems associated…

Let $(M^n,g)$ be a smooth compact Riemannian manifold with $Ric\geq 0$ and second fundamental form $\Pi\geq 1$ on its boundary $\Sigma$. Let $u\in C^{\infty}(M)$ be a positive solu…

Let $M^n$ be a compact Riemannian manifold with $Ric\geq 0$ and second fundamental form $\Pi\geq 1$ on $\partial M$. Let $u$ be a positive solution of … where $\lambda>0$ and…

Let $T_B(z,\phi)$ be the time map defined in the paper for parameters $z$ and $\phi$, with $z\in(0,2)$ and $\phi\in(0,\frac{\pi}{2})$. Convexity conjecture. For each…

Let $cOmega$ be a bounded domain, let $comegasubsetsubsetcOmega$ be the region where the nonnegative datum is supported, and let $u$ be the function given by Theorem 1 with…

Trudinger–Wang's conjecture. The first boundary value problem has a unique smooth solution $u$ if $\phi$ is affine mean convex. This conjecture concerns existence and uniqueness fo…

Bradlow vortex uniqueness conjecture. The only solution satisfying the Bradlow equation on $\mathbb{D}^2$ with $u(R)=0$ is the axially symmetric solution above, with all vortex pos…