23 problems

Let $\Omega$ be a domain for which the limit problem admits the second solution described in the paper, and let the corresponding finite-$p$ problem have mountain-pass solutions. M…

The $C^{1,\frac{1}{3}}$ regularity conjecture. Every such viscosity solution belongs to the Hölder space $C^{1,\frac{1}{3}}$.

Regularity conjecture. Infinity-harmonic functions are conjectured to belong to $C^{1,\frac 13}$.

Weighted p-Laplace convergence conjecture. The solution of the weighted $p$-Laplace equation converges to the solution of the infinity-Laplace equation as $p\to\infty$.

The $C^{1,\frac{1}{3}}$ regularity conjecture. Every such viscosity solution belongs to the class $C^{1,\frac{1}{3}}$.

Absolute maximum principle for tight forms. The restriction of $|F|$ to $\overline{U}$ attains its maximum on $\partial U$. This would extend the absolute-minimization theorem from…

Let $\mathbb{G}$ be a step two Carnot group with horizontal metric $g_h$. Denote by $\Sigma$ the horizontal layer in exponential coordinates, … Let $u$ be the fundamental solution…

Tangential-gradient conjecture. For infinity harmonic limits of $\mathrm{L}^p$-problems, $\nabla_\tau u$ coincides with $\nabla u$. The conjecture would identify the tangential-gra…

Stochastic-process convergence conjecture. The expected values of the stochastic process whose dynamic programming principle is modeled on $\bar{\mathcal{A}}^o_\varepsilon$ converg…

The infinity Laplacian is denoted by $\Delta_\infty$, and an infinity harmonic function is a function satisfying $\Delta_\infty u=0$ in the viscosity sense. For a domain in the pla…

Let $[-1,1]^2\setminus\{0\}$ be the punctured square, and let an infinity harmonic function on this domain be understood in the usual viscosity sense. A ground-state conjecture ass…

Let $u_\infty(x,y)=|x|^{4/3}-|y|^{4/3}$ be the Aronsson solution, let $\mathcal{O}$ be the square shown in the corresponding numerical experiment, and let $x_0,x_1,x_2,x_3$ denote…

Let $u(x,y)=e^{ix}-e^{iy}$, where $e^{it}=(\cos t,\sin t)$, be the vectorial Eikonal boundary datum, let $\mathcal{O}$ be a domain with $0\notin\mathcal{O}$, and let…

Let $u(x,y)=(x^2+y^2)^{1/2}$ be the scalar Eikonal solution, let $\mathcal{O}$ be a domain with $0\notin\mathcal{O}$, and let $\sigma^\mathcal{O}$ denote the limiting measure assoc…

Gehring-type conjecture. For each $\alpha>3/2$, there exists some $\epsilon_\alpha>0$ such that $\epsilon_\alpha\to0$ and

Let $\Omega$ be the domain under consideration, let $\widehat{v}$ be the maximal normalized solution of the infinity-eigenvalue problem, and let $\{u_p\}_{p>1}$ be the family of no…

$p$-harmonic approximation conjecture. The limiting mapping $u_\infty$ is a generalized solution of (1.1), in the sense of $D$-solutions, also in the full vectorial case.

Uniqueness conjecture. The good solution selected by the method of $L^p$ approximations as $p\to\infty$ should be unique.

Planar $C^{1,1/3}$ regularity conjecture. Every viscosity solution of the infinity-Laplace equation in the plane belongs to

Half-concavity conjecture. The solution $u$ is $(1/2)$-concave. Equivalently, the function $w$ is convex; formally, $w$ solves

Let $\alpha(d,p)$ denote the optimal universal Hölder continuity exponent for gradients of $p$-harmonic functions in dimension $d$, with $p\to\infty$. Asymptotic Hölder exponent co…

Starkness characterization conjecture. Starkness characterizes Aronsson maps: a map is Stark if and only if it is a solution of the full Aronsson PDE system.

Let $\Omega$ be the domain under consideration. Assume that $u,-v\in\operatorname{USC}(\bar\Omega)$ and $f\in C(\Omega)$ satisfy … Suppose that either $u$ has no strict local maxim…