33 problems

Let $(K,d,bc)$ be a compact metric space with Hausdorff measure $bc$, and let $(\mathcal{E},\mathcal{F})$ be a strongly local regular Dirichlet form whose semigroup has heat kernel…

Critical exponent conjecture. The critical value of $b1$ for the rigidity-flexibility dichotomy is one of

Let $\tilde{X}_{t}$ be the continuous version of the density at time $t$, and let $H(x)$ denote its optimal Hölder index at $x$. For $\eta$ in the relevant interval, let $D(\eta)$…

Let $\gamma_1$ and $\gamma_2$ be critical invariant circles in the same universality class, and let $h_{\gamma_i}$ be their conjugacies to the reference dynamics. Write…

Let $\Omega\subset\mathbb{R}^N$ be open and bounded with a $C^{1,1}$ boundary, let $p>1$ and $0<s<1$, and let $f\in L^q(\Omega)$. Let $u\in W^{s,p}_0(\Omega)$ solve the correspondi…

Concentrating-parameter criterion. It is necessary and sufficient for

De Giorgi's conjecture. The lower bound $\exp(-c_nK)$ is highly suboptimal: the optimal Hölder exponent should have algebraic dependence on $K$, similarly to the examples of Piccin…

Fix $\theta\in\mathbb{R}$, and let $utheta$ be the invariant measure associated with the solution $V^{(theta)}$ of the SPDE started from the constant initial profile $theta$. Assu…

Let $u$ be a suitably defined weak solution to the thin-film equation with exponent $n\geq 3$, and let $\operatorname{supp}u(\cdot,t)$ denote its spatial support at time $t$. Const…

Let $p\in(2,\infty)$, let $B_1$ be the unit ball, and let $\Delta_p u\coloneqq\operatorname{div}(|Du|^{p-2}Du)$. A function is locally of class $C^{1,\alpha}$ if it has locally Höl…

Optimal regularity conjecture. Any $K$-finite matrix coefficient of a unitary representation of $U$ is in $C^{(r,\delta)}(U_r)$, and this regularity is optimal.

Piccinini–Spagnolo Hölder exponent conjecture. In higher dimensions, the Hölder exponent should have the form

Let $d$ be the ambient dimension, let $1\leqslant m\leqslant d-1$, and let $\alpha$ be the Hölder exponent. Write $\mathbf{CH}^{m,\alpha}(\mathbf{R}^d)$ for the space of $\alpha$-f…

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…

Spectrum-equality conjecture. Even though necessarily $\alpha_{x_0}(t)\neq\alpha_0(t)$, one has

Let the Sierpiński gasket carry Brownian motion, with generator defined as the self-adjoint operator analogous to the Laplacian. A function is α-Hölder if its Hölder exponent is α.…

Kinetic integro-differential Krylov–Safonov conjecture. There exist constants $\alpha>0$ and $C$, depending only on the dimension, $\lambda$, and $\Lambda$, such that

Kinetic Krylov–Safonov conjecture. There exist constants $\alpha>0$ and $C$, depending only on the dimension and $\lambda,\Lambda$, such that

Let $f=(f^i)_{i=1}^k$ and $g=(g^i)_{i=1}^k$ be the data appearing in Theorems 1 and 2, and let $i,j\in\{1,\ldots,k\}$. A diagonal assumption on $g$ means the corresponding hypothes…

Let $\Gamma_\sigma$ denote the class of quasislits whose driving terms have the relevant bound $\sigma$, and let the Hölder exponent refer to capacity parametrization on intervals…

Hölder-exponent conservation conjecture. The Hölder exponent is conserved in time by the multilevel system given by the paper's equation, so that

For any $\delta>0$, let $\mathcal{C}_{\mathfrak{s};\mathrm{loc}}^{-\delta}(\mathbf{R}_{>0}\times\mathbf{T}^{2})$ and $\mathcal{C}^{-\delta}(\mathbf{T}^{2})$ denote the correspondin…

Exact exponent conjecture. There exists an event $\Omega_1$ of probability $1$ such that, for every $\omega\in\Omega_1$ and every non-degenerate compact interval…

Equality of Hölder exponents. One has

Sharpness conjecture. The Hölder exponent in Theorem … for the solution interpreted as an $\mathcal{H}$-valued process.