137 problems

For every integer $n\ge 3$ and every convex planar $n$-gon $P$ of area $A$, let $\lambda_1^N(P)$ denote the first nonzero eigenvalue of the Laplace operator with Neumann boundary c…

For every bounded simply connected planar domain $\Omega$ with infinite-mass boundary condition, every mass $m\ge 0$, and the disk $B_{|\Omega|}$ having the same area as $\Omega$,…

For every initial datum $u_0\in\dot H^{1/2}(\mathbb R^3)$ and every time interval $I$ containing $0$, if $u,v\in C(I;\dot H^{1/2}(\mathbb R^3))$ are solutions of the cubic nonlinea…

For every integer $d\ge 1$, every bounded open set $\Omega\subset\mathbb{R}^d$ of finite positive volume $|\Omega|$, and every $k\in\mathbb{N}$, if $\lambda_k(\Omega)$ denotes the…

Let $T$ be a nondegenerate planar triangle, and let $0<\lambda_1^N(T)\leq\lambda_2^N(T)$ be its first two positive Neumann Laplace eigenvalues. Define the harmonic and geometric me…

For the three-dimensional incompressible Navier–Stokes equations $\partial_t u+(u\cdot\nabla)u=-\nabla p+\nu\Delta u+f$, $\nabla\cdot u=0$, with smooth divergence-free initial data…

For every fixed integer $d\ge 1$, let $C_{\rm LY}(\beta,d)$ denote the optimal constant such that every positive solution $u$ of $\partial_tu+(-\Delta)^{\beta/2}u=0$ on…

The Schiffer conjecture asserts that if a bounded connected domain $\Omega\subset\mathbb{R}^n$ with sufficiently smooth boundary admits some $\lambda>0$ and a nonconstant function…

For each dimension $N\ge 2$ and higher order $m\ge 2$, determine the sharp constants and all extremal divergence-free vector fields in the corresponding higher-order Heisenberg unc…

For the two-dimensional stationary Prandtl system under a constant adverse pressure gradient, determine the local asymptotic structure of a solution near a finite separation point…

For the Fermi–Pasta–Ulam system in the standard continuum-limit scaling, determine whether solutions with initial data at $L^2$ regularity converge, in the appropriate norms, on ev…

Let $\varphi$ be a highest-wave profile for a Whitham-type equation, with crest at $x=0$ and crest regularity $\varphi\in C^{s}$ for some $s\in(0,1)$. The conjecture asserts that,…

Let $V_n$ be a sequence of divergence-free matrix fields with $\sup_n\|V_n\|_{L^1}<\infty$, each supported in an open set $A_n$ satisfying $|A_n|\to 0$. The conjecture asserts a co…

For every smooth magnetohydrostatic equilibrium $(\Omega,p,B)$ in Euclidean $3$-space, where $\Omega$ is an embedded solid torus, the regular level sets of the pressure $p$ are nes…

For the three-dimensional incompressible Navier–Stokes equations $\partial_t u+(u\cdot\nabla)u=-\nabla p+\nu\Delta u$, $\nabla\cdot u=0$, with $\nu>0$ and smooth divergence-free in…

Let $(M,g)$ be a smooth closed Riemannian manifold of dimension at least $3$, let $\Sigma\subset M$ be a fixed smooth submanifold of codimension $2$, and let $\phi_\lambda$ be an…

For fixed singular or big-cohomology data for which twisted Kähler–Einstein currents are defined, if $T_1$ and $T_2$ are two twisted Kähler–Einstein currents satisfying the same tw…

For every integer $n\ge 3$ and every area $A>0$, among all convex $n$-gons $P\subset\mathbb{R}^2$ with $|P|=A$, the regular $n$-gon $R_{n,A}$ uniquely minimizes the first Dirichlet…

For every integer $n\ge 3$, there exists a constant $C_n>0$ such that, for every nonzero real-valued function $u$ satisfying $\Delta u=0$ in $B_4\subset\mathbb{R}^n$ and $u(0)=0$,…

Let $\Omega\subset\mathbb{R}^d$ be a bounded $\mathrm{C}^{1,\alpha}$-domain, and let $A_{1,\sigma,n}$ denote the Stokes operator with no-slip boundary conditions on…

For every speed $c\in(0,\sqrt{2})$, there exists a nonconstant finite-energy profile $\psi:\mathbb{R}^2\to\mathbb{C}$ such that the traveling wave $\Psi(t,x)=\psi(x-ct e_1)$ solves…

For every integer $n\ge 2$ and every smooth uniformly elliptic parametric integrand $\Phi$ on $\mathbb{R}^{n+1}$, there exists an exponent $\alpha=\alpha(n,\Phi)>0$ such that every…

For every dimension $n\ge 2$ and every $p>2$, let $\Omega\subset\mathbb{R}^n$ be open and let $u\in W^{1,p}_{\mathrm{loc}}(\Omega)$ be a weak solution of the inhomogeneous $p$-Lapl…

Let $f_t$ be a sufficiently regular solution of the spatially homogeneous Landau equation with Coulomb interaction, and let…

Does there exist an integer $k\ge 1$ and a nonempty $C^k$-open family of smooth, autonomous, divergence-free velocity fields $\mathbf u\colon\mathbb T^3\to\mathbb R^3$ such that ev…