51 problems

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…

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 $\Lambda_n:= \sup_{U \in \mathcal{S}\mathcal{C}(\mathbb{R}^n)} \lambda_1$ be the supremum of the first stability eigenvalue over singular minimizing cones in $\mathbb{R}^n$, an…

Global asymptotic-shape conjecture. The rescaled flow $\hat\gamma(\cdot,t)$ converges as $t\to\infty$ exponentially fast in the smooth topology to a limit $\gamma_\infty$. Furtherm…

Critical-dimension conjecture. The critical dimension $d^*$ is $6$. In particular, for any minimizer $u$ of the Alt–Caffarelli problem in $\Omega\subset\mathbb{R}^d$ with $d\leq 6$…

Finite-time convergence conjecture. Then $u(t,\cdot)$ converges to $u_\phi$ in finite time.

For $q\geq 1$, consider the ellipsoid … Ellipsoid existence conjecture. There is a non-planar free boundary minimal immersed disk into this ellipsoid if and only if $q>3$. This is…

Let $u$ be a solution of the no-sign obstacle problem … and let $\Sigma_{n-1}$ denote the top stratum of the free boundary, with $\Sigma_{n-1,a}$ its anomalous part. Here…

Let $u$ be a global solution to the thin obstacle problem with cubic growth, and suppose that its positivity set on the thin space is bounded. Cubic positivity-set conjecture. The…

Let $\mathbb{S}^3_+$ denote the round hemisphere and let $\mathbb{H}^3$ denote hyperbolic three-space. A free boundary minimal annulus is a compact minimal annulus in a geodesic ba…

Let $d$ be the spatial dimension, let $\frac{d}{2}+1<\sigma<\infty$, and let $(v,B,\Gamma)\in C([0,T);\mathbf{H}^{\sigma})$ be a solution to the free boundary MHD equations. Write…

Let $(u,L)$ solve the free-boundary problem, and define … Let $\Pi_{\min}(x):=\int_x^{\infty}\pi_{\min}(y)\,dy$ denote the integrated minimal travelling wave. Minimal travelling-wa…

Let $\eta_0$ be the initial data, with initial domain ${\eta_0>0}$. Condition $(C)$ means that $\eta_0$ is strictly subharmonically ordered with $\nu$ with respect to ${\eta_0>0}$…

Let $l,m,k$ be parameters and let $\varphi:(\mathbb R_+)^l\times \mathbb R^{m}\to \mathbb R_{\geq 0}$ solve … in $(\mathbb R_+)^l\times \mathbb R^{m}$, with … on…

Let $S^1=\{y\in\mathbb{R}^2\mid y_1^2+y_2^2=1\}$, and let $\mathfrak{B}^\circ_\delta[f]$ and $\mathfrak{B}^{\circ,*}_\delta[f]$ denote, respectively, the minimal locally concave an…

Variational solution threshold conjecture. There exists a constant $bbda_{b1,D}$ with $bbda_{b1,D}>bbda_{b1,D}$ such that, for every $bbda a bbda{b1,D}$, the variational problem ha…

Ball solution-count hypothesis. There exists a constant $bbda_{b1,D}>0$ such that the problem has exactly two solutions for $bbda>bbda_{b1,D}$, exactly one solution for…

Interval solution-count conjecture. There exists a constant $bbda_{b1,D}>0$ such that the problem has exactly two solutions for $bbda>bbda_{b1,D}$, exactly one solution for…

Strong-convergence conjecture. The renormalized sequence converges strongly:

Consider the Signorini problem, also known as the thin obstacle problem, and its free boundary. Generic smoothness conjecture. Generically, the free boundary in the Signorini probl…

Let $u$ be a minimizer of the functional in on some open set $\Omega\subset\mathbb R^n$, and let $\Gamma_{\mathrm{BP}}(u)$ denote its branch-point set. For a relatively compact set…

Let $\mathbb{B}^3$ be the unit ball in Euclidean three-space, and let a free boundary minimal annulus be an embedded minimal annulus whose boundary lies on $\partial\mathbb{B}^3$ a…

Let $M$ denote the total myosin mass of a traveling wave solution and let $V$ denote its velocity. Consider parameter ranges in which $M$ decreases as a function of $V$ for small…

Let $\rho(z,\phi)$ be the relaxed functional for an admissible pair $(z,\phi)$, and define … If $\lambda^n(C\Phi)>0$, then $z$ is not uniquely identified and may attain every value…