157 problems

Consider a bounded regular domain $\Omega\subset\mathbb{R}^n$ and the divergence-form equation … Here the coefficients $a^{ij}$ are bounded and measurable, and $u$ is a local weak…

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…

Let $s_c$ be the scaling-critical Sobolev index of a partial differential equation, meaning that the homogeneous $H^{sc}$-norm is invariant under the equation's scaling symmetry.…

Christodoulou's conjecture. The initial value problem (1.1)--(1.5) is well posed in Sobolev spaces if

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 $(M,g)$ 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…

For the Sierpiński carpet, let $\alpha_1$ denote its critical Besov exponent, let $d_h$ be its Hausdorff dimension, and let $d_{th}$ be its topological Hausdorff dimension. Sierpiń…

Let $\Omega\subset\mathbb{R}^d$ be a bounded domain, let $\gamma$ be the conductivity, and assume $d\geq 3$. The Dirichlet-to-Neumann data determine $\gamma$ uniquely. Brown's conj…

Let $\mathbf W^r_{1,\alpha}$ be the weighted Sobolev-type function class, and let $\kappa_m(\mathbf W^r_{1,\alpha})$ denote the minimal cubature error over formulas with $m$ knots.…

Let $\Omega\subset\mathbb R^2$ be a domain and let $\varphi$ be a $K$-quasiconformal mapping, with $K\geq 1$. Let $\beta_0$ denote the exponent appearing in the generalized integra…

Higher-order equivalent-norm conjecture. The two displayed equivalent-norm estimates should also hold for every $r\geq 3$.

Optimality conjecture. When $d>1$, these rates are also optimal. The rate is known to be optimal for $d=1$ in several related models, whereas the source leaves the higher-dimension…

Wannebo's conjecture. The decompositions

Odd-order decomposition conjecture. The decomposition above holds for all odd $m$, $p>1$, $N$, and open $\Omega\subset\mathbb{R}^N$; for even $m$, there are $p$, $N$, and $\Omega$…

Quintic NLS well-posedness conjecture. Local and global well-posedness should hold in $H^s(\mathbb T)$ for all $s\geq 0$.

For $\alpha\in\mathbb{R}$, let $H^\alpha(\mathbb{R}^n)$ denote the Sobolev space, and let $f\in H^\alpha(\mathbb{R}^n)$ be the initial datum for the linear Schrödinger equation … w…

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

Let $n\in\mathbb{N}$ and $0\leqslant k\leqslant n-1$, and let $A^2_{n,k}(x)$ denote the squared norm of the derivative-evaluation functional on the closed interval under considerat…

Let $n\in\mathbb{N}$, let $k\in\{0,1,\ldots,n-1\}$, and let $A_{n,k}(x)$ be the norm of the evaluation functional $f\mapsto f^{(k)}(x)$ on the Sobolev space of functions satisfying…

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

Let $p$ and $\alpha$ be the parameters from the preceding theorem, let $\{\lambda_n\}$ be a sequence in $\big(0,\frac{1}{p^\alpha}\big)^\infty$ satisfying…

Let $(\mathcal{H}_{k_\varepsilon})_\vartheta$ denote the power space associated with the kernel $k_\varepsilon$, where $\vartheta>1$, and let $\tau$ be the Sobolev smoothness param…

Let $t=\cos(\beta\pi)$ with $\beta\in(0,1)$, let $n\geq 2$, and let $H^s(\mathbb{S}^n)$ be the usual Sobolev space. Write $M_t^{(1-n)/2}$ for the generalized Minkowski–Funk transfo…

Lambert's conjecture. The $1/2$-Sobolev regularity condition is sufficient for the relevant convergence to the Gaussian distribution if and only if $\beta\le 2$.

Reznikov's conjecture. For every $\epsilon>0$ there exists a constant $c>0$ such that