21 problems

Let $n\ge 2$, $1<p<2$, and let $\alpha_{\rm loc}(n,p)>0$ be an admissible interior Hölder exponent for gradients of local $p$-harmonic functions. Is it true that, for every…

Boundary regularity conjecture. For every $x\in[0,A]$, there exists a version of $s\mapsto p^{![2]}_{r,i,c,k}(x,s,l,j)$ that is continuous at $0$ and at $A$. Moreover, there exist…

Let $\Omega$ be the bounded domain in the nonlocal Poisson problem, let $u$ solve that problem, let $f\in L^2(\Omega)$, and let $\beta$ be the kernel exponent. Continuity conjectur…

Local Hölder regularity conjecture. Then $u$ is locally Hölder continuous in $\Omega$.

Let $f$ be a smooth uniformly bounded function, and let $\mathbb{U}$ and $\mathbb{V}$ be the two functions associated with $f$ in the paper's Bellman-function formulation.…

Let $f$ be a smooth uniformly bounded function, and let $\mathbb{B}_\varepsilon$ denote the minimal function associated with $f$ on the domain $\Omega_\varepsilon$. $C^{1,1}$ regul…

Mumford–Shah conjecture. If $(K,u)$ is a minimizer of $E_\lambda$, then $K$ has the structure described above.

Let $\Omega\subset\mathbb{R}^N$ be a domain and consider weak solutions of … Assume that the coefficient matrix satisfies … for $\xi\in\mathbb{R}^N$ and almost every $x\in\Omega$,…

Let $X$) be a compact Riemannian manifold of dimension $d$, and let $f\in C^{r+\alpha}(X)$, with the persistence quantity $\Pers_p(H_k(X,f))$ defined as the $p$-persistence of the…

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 $\varphi$ solve the complex Monge–Ampère equation considered in the paper, with right-hand side $F$. The question concerns estimates for the regularity of $\varphi$ under assum…

Let $u$ be a global solution of an equation of minimal-surface type on $\mathbb{R}^n$, corresponding to a functional $A_{\Psi}$. Here $B_r$ denotes the radius-$r$ ball, $\nabla u$…

In the setting of variational problems with a coercive conformally invariant Lagrangian of quadratic growth, consider its critical points. Hildebrand's regularity conjecture. Every…

Let ${\Bbb D}$ be the unit disk, and let $f_0:\overline{{\Bbb D}}\to\overline{{\Bbb D}}$ be a homeomorphism of finite distortion with … Here ${\cal A}:[1,\infty)\to[1,\infty)$ is c…

Let $\Omega$ be a bounded open domain in an $n$-dimensional Alexandrov space with curvature bounded from below by $k$, and let $N$ be a compact smooth Riemannian manifold. Suppose…

Let $d,n\in\mathbb{N}$ and $1<p<\infty$. Let $h\colon\Omega\to\mathbb{R}^n$ be $p$-harmonic on an open set $\Omega\subset\mathbb{R}^d$, and let $V$ denote the standard nonlinear tr…

Regularity via minors (odd $k$). If $\bigwedge^k df\in\widetilde H_{>k}$ is smooth, then $f$ is smooth.

Existence conjecture. There exists $1<\underline{m}<(p^*)'$ such that, if $f\in L^m(\Omega)$ with $m\geq\underline{m}$, then system $$ admits a weak solution $(u,\varphi)\in W^{1,p…

Let $N$ be the ambient dimension, let $q>N$, let $\rho\in L^q_{loc}(\mathbb{R}^N)\cap\mathcal{X}^*$, and let $u_\rho$ be the minimizer associated with $\rho$. Regularity conjecture…

Let $\lambda$ and $\Lambda$ be the ellipticity constants in Proposition 2.1, and let $\varepsilon$ denote the exponent in its $W^{2,\varepsilon}$ estimate. Optimal exponent conject…

Epsilon-regularity conjecture. There exist constants $\varepsilon,C,r_0>0$, depending only on $\Omega$, $J$, and $\sigma$, such that if