12 problems

1/3-Hölder gradient conjecture. Infinity-harmonic functions in the plane should have a $1/3$-Hölder continuous gradient.

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…

Residuality conjecture. Residuality with exponent $\alpha\in(2,3)$ refers to the Hölder continuity of the second derivative.

Let $\adaPG{}$ denote the adaptive proximal gradient method studied in the paper, and let accelerated algorithms refer to the accelerated methods considered there. Hidden Hölder gr…

Local Hölder continuity conjecture. In general, the joint spectral radius is locally Hölder continuous on $\mathcal{H}(d)$, with order

Locally uniform trajectory-bounds conjecture. There is a constant $\Theta_d>0$, depending only on $d$ and the norm, such that for every $\mathcal{M}\in\mathcal{H}(d)$ with…

Consider the periodic functions $P$ arising from the divide-and-conquer recurrence with $g(n)=0$ and $f(1)=1$, first with $(\alpha,\beta)=(\alpha,1)$ and then with…

Consider diagonal systems in the critical case $p=n=2$, with bounded weak solutions and the usual natural growth and ellipticity assumptions. The smallness condition $K_0M<\lambda$…

The uniqueness conjecture. Such a solution $u(t)$ cannot possess another solution $v(t)$ satisfying these initial conditions.

Let $X_t(x)$ denote the density solving the stochastic partial differential equation associated with the one-dimensional superprocess, with spatially correlated noise determined by…

Graversen's conjecture. Every $1/2$-Hölder continuous function is polar for planar Brownian motion.