222 problems

Let $\mathbb{B}^3$ be the unit ball in $\mathbb{R}^3$. A free boundary minimal annulus is a genus-zero free boundary minimal surface with two boundary components. Two such surfaces…

Let $\theta_{\rm w}^{\rm d}$ be the detachment angle, and let state (2) be the weak or strong uniform state satisfying the local shock conditions for a wedge angle…

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

Let an embedded annulus-type minimal Lagrangian surface with Legendrian capillary boundary be an embedded minimal Lagrangian surface in the unit ball $\mathbb B^4$ whose topology i…

Waiting-region threshold conjecture. There exists a function $\Gamma(x,t)$ such that

Let $Z>0$ and consider the families of traveling-wave solutions of model C bifurcating from the stationary solution at solutions $P_0$ of the bifurcation condition … ordered by inc…

Sharp asymptotic rate conjecture. As $t\to\infty$, for some positive constant $c_\alpha$ depending on $K(\cdot)$,

We consider periodic traveling waves at the free surface of a two-dimensional, infinitely deep, incompressible inviscid fluid of constant vorticity, under gravity and without surfa…

Consider the $(r,\mathcal S)$-plane, where $r$ is the Bernoulli constant and $\mathcal S$ is the flow force constant. The cuspidal region is the region bounded by the two curves re…

Let $f\in C_0^\infty(\mathbb{R})$ satisfy $f\ge0$. Let $u$ solve … and suppose there is a homeomorphism $\Phi:\mathbb{R}^n\to\mathbb{R}^n$ and a one-dimensional solution $U$ of the…

Let $(M^{n+1},\partial M)$ be a compact Riemannian manifold with smooth boundary. A free boundary minimal hypersurface (FBMH) is a smooth embedded hypersurface…

Consider a positive function $u$ homogeneous of degree one in a domain $E gsubseteqmathbb{R}^n$, where $partial E$ is a cone, solving the one-phase free boundary problem … Assume t…

Let $(u_\rho,\psi_{u_\rho})$ denote the stationary solutions constructed in Theorem 3, where the construction uses a sequence of parameters tending to infinity, and let $\lambda$ d…

Benjamin–Lighthill conjecture. The Bernoulli constant satisfies

RAPM two-parameter convergence conjecture. As either $C\to0^+$ or $R\to0^+$,

Let $D$ be an optimal configuration in a planar domain, and let $\partial D$ denote its free boundary. Two-dimensional free-boundary regularity conjecture. In dimension two, the fr…

Let $\Omega$ be a convex domain, and let $D=D_{\alpha,A}$ be an optimal configuration in $\Omega$. Write $D^c$ for its complement and let $\overline{\alpha}_\Omega(A)$ denote the t…

Let $(u,K)$ be a minimizer of the Mumford–Shah functional in dimension $n=3$, and call $K$ its singular set. Three-dimensional Mumford–Shah singular-set conjecture. The singular se…

Electric-field insufficiency conjecture. When $cmu/(aQ)>cpi^2/2$, the electric field is too weak to prevent breakthrough caused by the hydrodynamic dipole. The claim concerns the a…

Let $N=2$, let $p\in(1,p_N)$, and assume that the non-degenerate bound … holds for positive constants $C_1,C_2$ and a function $\tilde g$ satisfying the stated assumptions. Let…

Waiting-time conjecture. When $\alpha=1$, the boundary behavior $c\sim x$ should, after some time, cease to persist; the boundary should then start moving and attain the physical v…

Mesa limit and uniform projection conjecture. The flows should converge to a Hele–Shaw or mesa-type free-boundary evolution whose terminal state is the Wasserstein projection of…

Let $N\geq 2$, $1<p<p_N$, and let $\Omega$ be a two-dimensional domain symmetric and convex with respect to the coordinate directions. Consider the global simple curve of unique so…

Let $C$ be a convex set that is not a ball, let $g$ be a generic non-constant function on $\partial C$, and let $\mathcal{QS}(f,g)$ denote the corresponding overdetermined free bou…

Let $g$ be the reduced value function, let $w_0$ and $b^*$ be switching points with $w_0<b^*$, let $K_1,K_2,K_4$ be constants, and let $v$ be the function appearing in the middle-r…