23 problems

Sobolev regularity conjecture. For $\beta \in (0,8\pi)$,

Let $s>1$ and let $\gamma^*$ denote the supremal Sobolev regularity of the target potential and initial density, … Assume that $\max\{d/2,s-1\}\leq\gamma^*<\infty$ and that the qua…

Let $\Omega\subset\mathbb{R}^3$ be a bounded domain with boundary $\partial\Omega$, and let $D\subset\partial\Omega$ be a buried boundary part such that $\Omega$ is a proper subset…

The authors' conjecture. The observed power law is influenced by the limited small-scale separation accessible in the computations and would not occur under the assumptions conside…

Let $\Omega\subset\mathbb{C}^n$ be a smooth bounded pseudoconvex domain, let $N_q$ denote its $\overline{\partial}$-Neumann operator, and let the Diederich–Fornæss index of…

Let $s$ denote the Sobolev regularity exponent in Theorem. Sharpness conjecture for the local well-posedness Sobolev range. The range of $s$ in Theorem is sharp for generic problem…

Let $m$ be the order of the polyharmonic operator, whose principal part is $(-\Delta)^m$, and let $s$ denote the Sobolev regularity order of its coefficients. Optimal-regularity co…

Let $\Omega$ and $\Omega'$ be domains in $\mathbb{R}^n$ with $n\geq 3$, let $f_0\colon\Omega\to\Omega'$ be a given homeomorphism, and let…

Let $\Omega$ be the domain, let $f\in L^1(\Omega)$ be nonnegative, and let $u$ be a solution of the singular elliptic problem … with homogeneous Dirichlet boundary condition $u=0$…

Optimal-threshold conjecture. For $p>1$, this threshold should be optimal for obtaining solutions in $W_0^{1,p}(\Omega)$ for positive data $f\in L^m(\Omega)$, $m>1$.

Let $\rho$ be a passive scalar transported by a Sobolev vector field $u$, with initial datum $\rho_0$, and let $p$, $q$, and $r$ be the integrability exponents appearing in the pre…

Extension conjecture. The sufficient regularity theorem is also valid when $p=2$: every viscosity supersolution of

Let $G=B(0,R)\subset\mathbb{R}^3$, let $S$ be the sphere of transport directions, and let $I^\circ$ denote the interior of the energy interval. Consider the solution…

Let $\mathcal G_s(K)$ be the class defined in the paper by, and let $\mathcal F_s(g)$ be the class defined by. For a map $f$, write $K_f$ and $K_{f^{-1}}$ for the distortions of…

Conjecture. The gradient magnitude satisfies

Modena–Székelyhidi threshold conjecture. The threshold in the existing convex-integration constructions is not optimal; rather, the optimal threshold should be

Gehring-type conjecture. For each $\alpha>3/2$, there exists some $\epsilon_\alpha>0$ such that $\epsilon_\alpha\to0$ and

Let $u$ be a planar $\infty$-harmonic function. The preceding results establish Sobolev regularity for powers of $|Du|$ and measure bounds for the distributional determinant, motiv…

Conjecture on generalised minima. If, for every $\eta\in\mathbb{R}^{n}\setminus\{0\}$, $f_{\eta}$ is a strictly convex norm and generalised minima are unique modulo rigid deformati…

Let $0<s<2$ and $1<p<\infty$ satisfy $s<\frac{2}{p}$. Let $\mu,\nu\in F^s_{p,q}\cap L^\infty_c$ satisfy the ellipticity condition, and let $f$ be the principal solution to the Belt…

Let $\omega_0\in H^1(\mathbb{T}^2)\cap L^\infty(\mathbb{T}^2)$ be mean-zero, and let the corresponding solution of the two-dimensional incompressible Euler equations be a Yudovich…

Let $s\in(0,1)$ and let $\Omega=[-1,1]^2\setminus[0,1]^2$ be the L-shaped domain. Consider the fractional Dirichlet equation … A solution may be viewed together with the eigenfunct…

$L^2$-conjecture. The optimal regularity for the initial data set is