88 problems

Let $Q(g,f)$ denote the Boltzmann collision operator without angular cutoff, and let $s$ be the exponent determined by the angular singularity. For a fixed function $g$, write…

Let $\mu_d(v)=(2\pi)^{-d/2}e^{-|v|^2/2}$ be the Maxwellian equilibrium distribution on $\mathbb{R}^d$, let $Q$ denote the Boltzmann collision operator, and let $s$ with $0<s<1$ be…

Let $s\in(0,1)$, and consider the fractional Laplacian associated with a censored stable process on a convex domain. The best Hardy constant is the largest constant for which the c…

Mass-growth conjecture. Analogous mass-growth and boundedness results should hold at least for the $\alpha$-stable operator (fractional Laplacian) $\mathcal L=(-\Delta)^{\alpha/2}$…

Classification conjecture. For $\beta=(d+\alpha)/(d-\alpha)-1$, the functions $u_{c,A}$ constitute all bounded solutions of the stationary equation, while for…

Uniqueness conjecture. Every positive solution of the single-pole problem is, up to a rescaling, one of the functions $\varphi^\lambda_\mu$; these rescalings remain solutions and m…

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

Let $\Omega$ and the given function $z$ be smooth as in Theorem, and let $u$ be a weak solution of the Dirichlet problem. The nonlocal Neumann datum $\mathcal N_s u$ is conjectured…

Fractional De Giorgi conjecture. In sufficiently small dimensions $n$, every such solution $u$ in $\mathbb{R}^{n}$ is one-dimensional. This is the fractional analogue of the classi…

Critical-exponent duality conjecture. One should have

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 $k\geq 2$ and let $A$ be a $\lambda_k$-minimizer. Non-simplicity conjecture. The minimizing eigenvalue $\lambda_k(A)$ is not simple. This is presented as a consequence of the p…

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…

Let the minimization problem be posed on a union of $k$ copies of ${\mathbb R}^n$, and let a stable configuration consist of disconnected pieces in these copies. Separated congruen…

Let $K_\alpha$ be the kernel of the resolvent operator $(D^\alpha+2)^{-1}$ on $\mathbb R$, with $\alpha\in(2,4)$. Le–Pelinovsky conjecture. The kernel $K_\alpha$ is sign-changing f…

Let $\Omega$ be a domain, let $s\in(0,1)$, and let $u$ be the solution considered in Theorem, so that the quotient $u/{\rm d}_\Omega^s$ has boundary Hölder regularity. For $p=2$, o…

Let $\Omega\subset\mathbb{R}^N$ be a domain, let $s\in(0,1)$ and $p>1$, and let $u$ solve … with $f\in L^\infty(\Omega)$. Higher interior Hölder regularity conjecture. The solution…

Let $K$ be the Sierpiński gasket, let $d$ be its metric, let $d_h$ and $d_w$ denote its Hausdorff and walk dimensions, respectively, and let $G_s$ be the Riesz kernel associated wi…

Let $s$ denote the fractional order, with $0<s<1$, and consider the spectra of two domains for the corresponding fractional Laplacian. Fractional-order spectral distinction conject…

Let two domains be such that their spectra for the Laplacian are identical. Fractional Laplacian spectral distinction conjecture. Their spectra for the fractional Laplacian would n…

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…

Consider the fractional Anderson operator … on $\mathbb{Z}^d$, with $0<\alpha<1$, where $\lambda>0$ is the disorder parameter and the random potential has i.i.d. compactly supporte…

Proof-artifact conjecture. The extra $+2$ term in the condition is an artifact of the proof of the cited theorem; equivalently, the result should hold without that additional term.