26 problems

Ball's sufficient conditions conjecture. Strict positivity of the second variation, together with suitable versions of quasiconvexity in the interior and at the boundary, should im…

Let $\Omega\subset\mathbb{R}^N$ be open and bounded with a $C^{1,1}$ boundary, let $p>1$ and $0<s<1$, and let $f\in L^q(\Omega)$. Let $u\in W^{s,p}_0(\Omega)$ solve the correspondi…

Optimal growth rate conjecture. The optimal growth rate estimate given in Theorem holds if and only if $\gamma<2$ (thus $\theta<\frac{3\pi}{2}$).

Boundary regularity conjecture. At least in smooth domains $\Omega\subset\mathbb{R}^n$, one has

Let $\Omega$ be a domain, let $s\in(0,1)$, and let $u$ be the solution considered in Theorem, so that the quotient $u/{\rm d}_\Omega^s$ has boundary Hölder regularity. For $p=2$, o…

Let $\Omega\subset\mathbb{R}^2$ be a Jordan domain, and let $\partial\Omega$ denote its boundary. For $x\in\partial\Omega$, let $\varepsilon(x,r)$ be the Carleson $\varepsilon$-fun…

Let $\Omega\subset\mathbb R^n$ be a one-sided NTA domain whose Green function with some pole $p$ satisfies, for some $\delta>0$ and constants $c_1,c_2>0$, bounds of the form … for…

Let $n=2$. Let $\Omega$ be a one-sided NTA domain, and let $u$ and $v$ be as above, with $u/v$ agreeing on $B(z,r)\cap\partial\Omega$ with a real-analytic function $R$. Assume more…

Let $\Omega$ be a one-sided NTA domain, and let $u$ and $v$ be positive harmonic functions as above, vanishing on the relevant portion of the boundary, such that their quotient agr…

Let $G=\{x\in U:r(x)>0\}$ be a domain with $r\in C^1(U)$, smooth noncharacteristic boundary away from the grazing set, and suppose the transport problem … has homogeneous inflow da…

Boundary-lifting regularity conjecture. Less regularity of $g_N$ should still be sufficient to infer

Boundary regularity conjecture. The diffeomorphic minimiser of Dirichlet energy, known to have a $\mathscr{C}^{1,\alpha'}$ extension up to the boundary, is also a diffeomorphism on…

Let $u\in W^{2,n}_{\operatorname{loc}}(Q^+_1)\cap C(\overline{Q^+_1})$ solve … with $u=0$ on $T_1$, where the coefficients satisfy the stated measurability and uniform ellipticity…

Let $T$ be an $n$-dimensional absolutely area-minimizing integer rectifiable current in an open subset of $\mathbb{R}^{n+1}$ containing the origin. Near the origin, suppose that…

Let $D$ be a domain, let $t\in\mathbb R$, and let $\rho^D(t,\mathord\cdot)$ denote the density from Theorem 2.7. Boundary regularity conjecture. For each $t\in\mathbb R$, the funct…

Meyers' conjecture. Condition (ii) can be dropped if $\partial\Omega$ is smooth enough or if a smooth enough function is prescribed on the boundary as the datum.

Let $1<p<2$, $K>0$, and $0<q<1/p$, and set … For these parameters, the paper proves that there is a traditional barrier at $(0,0)$ for $\Theta$, although $(0,0)$ is irregular. Sing…

Let $K>0$, $q>0$, $1<p<\infty$, and let … For $p>2$, earlier Petrovski-type criteria give regularity of $(0,0)$ with respect to $\Theta$ when … where $\lambda=n(p-2)+p$. Lindqvist'…

Let $u$ solve the fractional $p$-Laplacian evolution equation in a domain $\Omega$, with homogeneous Dirichlet data outside $\Omega$, and let…

Assume the same capillary-surface setting with $n=2$, where $\Omega$ has a corner at $P$ of angular size $2\alpha$ and $\gamma(x)\to\gamma_{0}$ as $x\to P$ along $\partial\Omega$.…

Assume $n=2$. Let $\kappa\ge 0$, let $\Omega\subset\mathbb{R}^{2}$ have a corner at $P\in\partial\Omega$ of angular size $2\alpha$, where $\alpha\in\left(0,\frac{\pi}{2}\right)$, a…

Let $D,D'\subset\mathbb C^n$ be bounded domains with real analytic boundaries, and let $f:D\to D'$ be a proper holomorphic map. Holomorphic extension conjecture. The map $f$ extend…

The system under consideration is posed on a bounded domain $\Omega\subset\mathbb{R}^2$ with boundary $\partial\Omega$. The preceding results are established under the assumption t…

Let $g\left(e^{i\varphi},\theta\right)$ be a function in $C\left(\mathbb{S}^{1}\times\mathbb{S}^{d-1}/\mathbb{Z}_{2}\right)$ satisfying the brothers Riesz conditions. Let…

Boundary-vortex energy conjecture. The energetic cost of a boundary vortex remains of order