87 problems

For the three-dimensional axially symmetric steady incompressible cavity-flow free-boundary problem, determine whether the cavity solutions whose existence was established by Garab…

For every bounded simply-connected domain $\Omega\subset\mathbb{R}^2$, let $u$ be its torsion function, defined by $-\Delta u=1$ in $\Omega$ and $u=0$ on $\partial\Omega$, and let…

Let $B_1\subset\mathbb{R}^2$ be the unit disk, let $\varepsilon>0$, and define the Ginzburg--Landau energy…

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain, and let $\lambda_k(\Omega)$ and $\mu_k(\Omega)$ denote its $k$-th Dirichlet and Neumann Laplacian eigenvalues, respectively.…

For the incompressible Euler equations on boundary-free three-dimensional space, $\partial_t u+(u\cdot\nabla)u=-\nabla p$, $\nabla\cdot u=0$ on $\mathbb{R}^3\times[0,\infty)$, with…

For every admissible integer $n\ge 12$ and every exponent $p$ satisfying $p_{\mathrm{JL}}(n)<p<p_{\mathrm{JL}}(n-1)$, every positive smooth entire solution…

Let $\gamma$ and $\delta$ be real analytic functions with $\gamma_4\neq0$, let $k$ and $\ell$ be integers, and let $\mathcal{E}^{\gamma,\delta}_{k,\ell}$ denote the partial differe…

On blowup for large data. For initial data with sufficiently large energy, the solutions of equation blow up in finite time in the sense that the derivative $u_r(t,0)$ diverges as…

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$, and let $\lambda_k(\Omega)$ and $\mu_k(\Omega)$ denote its Dirichlet and Neumann eigenvalues, respectively. For positive integ…

Let $\mathcal{U}$ be an ultradifferentiable structure that is closed under derivation and invariant under composition with mappings of class $\mathcal{U}$. The Kotake-Narasimhan th…

Hypoellipticity characterization. The operator $L$ is locally $C^\infty$-hypoelliptic if and only if both $L_1$ and $L_2$ satisfy the superlogarithmic estimate.

Let $p\ge 2$ and let $\phi:U\to\mathbb{C}$ be a complex-valued $(p,1)$-harmonic morphism defined locally on the standard Euclidean space $\mathbb{R}^{2p-1}$. The nonexistence conje…

Let $\Omega$ be the domain, let $B_n$ be the normal-state magnetic field, and let Proposition provide the conditions under consideration. For a compact set…

Let $g$ satisfy the self-similar boundary value problem referred to as, with the boundary conditions described in the preceding discussion, including $g'(0)<0$. Nonexistence conjec…

Unique-solvability conjecture. The boundary value problem is uniquely solvable if and only if $\mathcal{N}^\Omega_\zeta=\emptyset$.

Cubic nilpotent-Jacobian conjecture. If $JH(z)$ is nilpotent, then $N_t(z)$ is a polynomial solution in both $t$ and $z$. This is presented as an equivalent formulation of the Jaco…

Consider the scalar conservation law … Let $u$ solve the fully nonlinear fourth-order approximation … where $f$ is strictly convex, $g$ is concave, and $g(0)=0$. Strong convergence…

Let $P(D)$ be a hypoelliptic partial differential operator on $\mathbb{R}^d$ which is orthogonally degenerate on its linear characteristic cone … Moreover, let $F_1\subset F_2$ be…

Let $v$ be a vector field on $\mathbb R^{3}$, with pressure $p$, satisfying the stationary Navier–Stokes system … where $\nu>0$, together with … and…

Schiffer conjecture. If this problem admits a non-trivial solution, then $\operatorname{\Omega_S}$ is a ball. The conjecture is a well-known open problem in geometric analysis and,…

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain, let $\nu$ denote the unit outward normal on $\partial\Omega$, and let $\beta>0$. Consider the overdetermined system ……

Let $\Omega$ be the bounded domain in the nonlocal Poisson problem, let $u$ solve that problem, let $f\in L^2(\Omega)$, and let $\beta$ be the kernel exponent. Continuity conjectur…

Chen–Véron conjecture. In dimensions $1$ and $2$, such a fundamental solution exists, is radial and locally integrable, and satisfies the displayed bound.

Let $\alpha(\{\delta_j\}_{j=1}^n)$ be the exponent associated with the multi-level logarithmically improved criteria, and let $s$ denote the Sobolev regularity exponent. Potential…

Strictness characterization.