52 problems

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, let $1<p<\infty$, and let $u\in C(\Omega)\cap W^{1,p}(\Omega)$ be continuous and Lebesgue monotone. Determine whether $u$ can…

For $n\geq 2$ and $m\geq 1$, let $u\in W^{1,n}(\mathbb{B}^n,\mathbb{R}^m)$ be a weak solution of…

For $p\text{ in }[1,\text{infinity})$, consider the Jacobian operator … where $\mathscr{H}^p(\mathbb{R}^n)$ is the real Hardy space and $\dot{W}^{1,np}(\mathbb{R}^n,\mathbb{R}^n)$…

Generic flow-map non-precompactness conjecture. For a residual subset of $\mathcal{U}_p$, the associated flow maps $\{\Phi_t\}_{t\geq 0}$ are not precompact in $L^1$.

Let $\textnormal{r}>2$ be such that the Lipschitz continuity conjecture holds, set $\mathcal{X}=W^{1,\textnormal{r}}(\Omega)$, fix $b\in\mathcal{X}$, and define … Let…

Small-data critical well-posedness conjecture. The Cauchy problem is well-posed for small data in $\dot H^{\frac{n}{2}}$.

Consider the cubic nonlinear Schrödinger equation referred to as, with initial data in the Sobolev space $H^s$. Local well-posedness conjecture. The cubic NLS equation is locally w…

Non-Abelian gauge-fixing conjecture. There exists a gauge transformation $\lambda\in W^{1,p}(\operatorname{Ad}P)$ such that

Let $cuOmega\textbackslashsubset\textbackslashmathbb R^n$ be a domain, let $K\textbackslashge 1$, and let $p\textbackslashge 1$. A mapping…

Weight-dependence conjecture. There exists a constant $C=C_G>0$ depending only on $G$ such that

Let $p_1,p_2\in(1,\infty)$, let $\mathscr{F}_{p_1}$ and $\mathscr{F}_{p_2}$ be the corresponding Sobolev spaces, and let $U_{p_1,+}$ and $U_{p_2,+}$ be their optimal potential func…

Let $(M,g)$ be a simple Riemannian manifold with boundary defining function $\rho$, and fix $\gamma>-1$. An $\alpha$-boundary defining function ($\alpha$-bdf) for $\partial_+SM$ is…

Optimal growth conjecture. There exists $C(s)>0$ such that, for every $t\in[0,t_0)$ and every $s\in(0,\sigma]$,

Strong-convergence conjecture. The renormalized sequence converges strongly:

Consider the cubic problems denoted by … . Sharpness conjecture. Generically, the problem … is ill-posed in $H^s$ for $s<2$. The theorem establishes local well-posedness above thes…

Let $u$ solve the cubic wave equation on $[0,\infty)$ with initial data $(u_0,u_1)\in\mathcal{H}^{1/2}\cap\mathcal{H}^1$, where $\boldsymbol{u}(t)=(u(t),\partial_tu(t))$ and…

The test-space structure conjecture. The test space $\mathcal{Y}$ is isometrically isomorphic to $H^1(\vec{b},\Omega)$.

Let $K\colon \mathbb{R}^d\to\mathbb{R}^d$ be homogeneous of order $1-d$ and smooth outside the origin. Let $\nu$ be a measure satisfying … For $f\in C_0^\infty(\mathbb{R}^d)$, cons…

Non-uniqueness conjecture. Under the stated integrability condition and suitable assumptions on $\beta$ and $T$, such a density $\rho$ and incompressible Sobolev vector field $u$ e…

Equivalence of the optimal fractional Poincaré constants. For $t>s\geq 0$, the optimal Poincaré constant is

Let $X( \Omega)$ be a Banach function space, and let the difference quotient criterion be the condition in which the estimate referenced as equation DQC1:1 characterizes membership…

Global uniform smoothness conjecture. Identifying $\mathbf S_U^N$ and $\mathbf M_U^N$, the soliton addition and removal maps are uniformly smooth globally,

Uniform smoothness conjecture. The maps $B^N_+$ and $B^N_-$ are uniformly smooth on bounded sets in $H^s$.

Fix $p\in[1,+\infty)$. Let $b\in C^{\infty}([0,T]\times\mathbb{T}^d)$ be divergence free and let $u_0\in W^{1,2}(\mathbb{T}^d)$. If $u^\nu$ is a solution of the advection diffusion…

Let $b\in L^{\infty}([0,\infty),W^{1,1}(\mathbb{T}^d,\mathbb{R}^d))$ be divergence free, and let $\rho$ denote the mixing rate associated with an initial datum $u_0$. Bressan's mix…