49 problems

Let $\Omega$ be a Lipschitz domain, and let $u\in H^1$ be a weak solution to the Dirichlet problem … Suppose that the set…

Iversen-type conjecture. If the critical set of $f$ is nowhere dense and the cluster set of $f$ has empty interior, then every cluster point of $f$ is an asymptotic value. In parti…

Converse Liouville property conjecture. Any amenable measured groupoid is Liouville.

Let $B\subset\mathbb{R}^{n}$ be a unit ball with $n\geq 3$. Let $u:4B\to\mathbb{R}$ be harmonic and satisfy $u(0)=0$, and let $N_u(0,\frac{1}{2})$ denote its doubling index at the…

Let $n\geq 3$, let $\mathbb{B}^n\subset\mathbb{R}^n$ be the unit ball with boundary sphere $\mathbb{S}^{n-1}$, and let $u\in C^{\infty}(\mathbb{B}^n)$ be positive and satisfy … whe…

Let $(M^n,g)$, with $n\geq 3$, be a compact Riemannian manifold with nonnegative Ricci curvature, and let the second fundamental form satisfy $\Pi\geq 1$ on $\partial M$. Let…

Let $M$ be a complete noncompact (open) Riemannian manifold with nonnegative Ricci curvature. For a fixed growth rate, consider the vector space of harmonic functions on $M$ with p…

Fixed-point conjecture. If for any $m>0$ we have $B_mf=f$, then $f$ is harmonic.

Let $u$ be a harmonic function in $B(0,1)\subset\mathbb{R}^n$ with $u(0)=0$, and let $Z(u)$ denote its nodal set. Nadirashvili's lower-bound conjecture. There exists a universal co…

Guo–Wang's conjecture. If $1<q<\frac{n}{n-2}$ and $0<\lambda\leq\frac{1}{q-1}$, then $u$ is constant.

Let $D$ be a bounded domain in $\mathbb{R}^m$, with $m\geqslant 2$. Let $A_x(r_1,r_2)$ be an admissible annular domain centred at $x$, with $r_2>r_1>0$, and let $B_{r_2}(x)$ be adm…

Let $B\subset\mathbb{R}^n$ be a unit ball, with $n\geq 3$, and let $u$ be harmonic in $4B$, vanish at the center of $B$, have stable growth in $B$, and satisfy…

Let $n=2$. Let $\Omega$ be a one-sided NTA domain, and let $u$ and $v$ be as above, with $u/v$ agreeing on $B(z,r)\cap\partial\Omega$ with a real-analytic function $R$. Assume…

Let $n=2$. Let $\Omega$ be a one-sided NTA domain, and let $u$ and $v$ be as above, with $u/v$ agreeing on $B(z,r)\cap\partial\Omega$ with a real-analytic function $R$. Assume more…

Let $\Omega$ be a one-sided NTA domain, and let $u$ and $v$ be positive harmonic functions as above, vanishing on the relevant portion of the boundary, such that their quotient agr…

Let $u$ and $v$ be as in Theorem, with $v$ the distribution associated to $u$ and supported on the singular set $\Gamma$. For a test function $\phi\in C_c^\infty(B_2)$, write…

Let $\Omega\subset\mathbb{C}$ be a bounded domain whose boundary consists of finitely many non-intersecting Jordan curves. For each polynomial boundary datum on $\partial\Omega$, c…

Let $\Omega\subset\mathbb{R}^n$ be a Lipschitz domain and let $\Sigma\subset\partial\Omega$ be relatively open with respect to $\partial\Omega$. Let $u$ be harmonic in $\Omega$ and…

Let $n=2$ or $n\geq 4$, let $u$ satisfy the hypotheses of Theorem (i), and let $x\in\mathbb{B}_n$. Schwarz–Pick conjecture. One has … This proposed refinement combines the sharp gr…

Let $(M^{n},g)$ be a smooth compact Riemannian manifold with boundary $\Sigma$, with $\operatorname{Ric}\geq 0$ and principal curvature tensor $\Pi\geq 1$ on $\Sigma$. Let…

Uniqueness conjecture. If $1<q<n/(n-2)$ and $0<a\leq 1/(q-1)$, then $u$ is constant.

Let $M^n$ be a complete Riemannian manifold with nonnegative Ricci curvature and maximal volume growth, and let $\mathscr{F}_u(r)$ denote the frequency function of a harmonic funct…

Harmonic-function dimension conjecture. For every $k\geq 0$,

Finite-dimensionality characterization conjecture. The group $G$ has polynomial growth if and only if $$ is finite dimensional.

Veech's conjecture. Every non-negative, $r$-median function on $D$ is harmonic.