98 problems

A divergence-form elliptic operator has bounded measurable coefficients satisfying a Carleson measure condition of the type arising from changes of variables in smooth elliptic or…

Lazer–McKenna conjecture. There should be at least $2k$ solutions of the boundary-value problem.

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 $(M^n,g)$, 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…

Consider the second-order boundary-value problem … with … where $K$ and $A$ are positive constants and $L(x)=a_0^2+(a_1^2-a_0^2)x$, with $a_0,a_1>0$. Exact multiplicity conjecture.…

Let $\mathcal{L}$ be an elliptic operator of order $2m$ on an open domain $\Omega\subset\mathbb{R}^N$, with $N\geq2$. For an open portion $\Gamma$ of $\partial\Omega$, impose homog…

The dynamical spherical MIT bag has a moving boundary, and the fermion field satisfies the MIT bag boundary conditions at that boundary. In the massive case, the field cannot be wr…

Let $L$ be a differential operator with bounded measurable coefficients on a domain $\Omega$, and let $G$ be its Green function. Suppose the associated transformed kernel has the q…

Let $D$ be a bounded domain in ${\mathbb R}^n$ with boundary of class $C^{1/p}$, where $0<p\leq 1$. Let $G(f)$ and $N(f)$ denote the solutions of the Dirichlet and Neumann problems…

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 $\mathcal{N}^\Omega_\zeta=\emptyset$.

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…

Let $\Gamma\subset\mathbb{R}^3$ be a boundary consisting of two components $\Gamma_1$ and $\Gamma_2$, each a plane convex curve, with total boundary curvature at most $4\pi$. Assum…

Let the limit objects arising from bubbling at the boundary be anti-self-dual instantons on the half space. Energy quantization conjecture. There is an energy quantization for thes…

Let $Y$ be a compact oriented $3$-manifold with boundary $\partial Y=\Sigma$, and equip $S^1\times Y$ with a product metric compatible with the boundary identification. Let…

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.

Conformal extremal-metric conjecture. There always exists a solution of the extremal equation satisfying this Dirichlet boundary condition, with the solution metric pointwise confo…

The paper considers systems such as (1) and (2), for which weak solutions to a Guderley–Morawetz problem are obtained on suitable domains by the methods developed there. Extension…

Let $R>0$, let $k\geq 1$ be an integer, and consider the radial problem at the resonant value $\lambda=\lambda_k^{\mathrm{rad}}$. Let $\mu_k^\star(\lambda)$ denote the threshold pa…

Let the ambient space be $\mathbb{R}^{d+1}$, with absorbing boundary identified with $\mathbb{R}^d$. Write $\boldsymbol{r}$ for the displacement along the boundary, let $\lambda>0$…

Let $T>0$, let $I=[-T,T]$, and let $s_0$ range over $[0,T]$. For $s_0\in[0,T]\setminus\mathbb{N}$, let $\lambda^{s_0}$ be the first eigenvalue of Problem … with $m=0$. First-eigenv…

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…

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…