13 problems

Let $\kappa(\gamma)$ denote the limiting curvature constant associated with the fractional Laplacian parameter $\gamma$, as defined in the surrounding results. Positivity conjectur…

Let $x=(x_1,\ldots,x_d)$ with $x_i\in{\mathbb R}^n$, signed masses $m(x_i)$ satisfying … and define … A configuration is stationary and stable when its first variation vanishes and…

Let $k\geq 2$, let $A$ be a $\lambda_k$-minimizer, and let $u_k$ be the corresponding eigenfunction. Assume that $\lambda_k$ is simple. Ball-support conjecture. For every copy…

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…

Nonlocal De Giorgi conjecture. Is $u$ one-dimensional, at least under suitable restrictions on $\alpha$ and $n$? This is the fractional analogue of De Giorgi's one-dimensional symm…

Interior $L^p$ estimate conjecture. One should have

Let $a\in(-1,1)$ and let $u$ be a bounded, smooth solution of the generalized fractional Allen–Cahn equation … Assume that $\partial_{x_n}u>0$. Generalized fractional De Giorgi con…

Uniform condition-number conjecture. There exist $s<0$ and a constant $C$ such that

Let $\theta_0$ satisfy the hypotheses of the Hölder $1/2$ a priori estimate conjecture, and let $\theta$ be a vanishing-viscosity limit of solutions of … Here…

Let $d\geq 2$, let $D\subset\mathbb{R}^d$ be an arbitrary bounded convex set, and let $\varphi$ be the solution of the fractional Dirichlet problem … with the boundary condition fr…

Let $d\geq 3$, and let $D\subset\mathbb{R}^d$ be an arbitrary bounded convex set. Let $\varphi$ denote the solution of the fractional Dirichlet problem … with the boundary conditio…