63 problems

Reuleaux triangle maximality conjecture. The Reuleaux triangle maximizes the first eigenvalue of the $p$-Laplacian among all bodies of the same constant width.

Consider the radial Neumann problem discussed above in dimension $N=1$, with $R=1$, $p=2$, and nonlinearity $g$. For the power nonlinearity $g(s)=s^{q-1}$, the bifurcation branches…

Let $\Omega\subset\mathbb R^2$ be a domain and let $\varphi$ be a $K$-quasiconformal mapping, with $K\geq 1$. Let $\beta_0$ denote the exponent appearing in the generalized integra…

Let $p\geq 2$ and let $p'=\frac{p}{p-1}$ be the Hölder conjugate exponent, so that … For functions with bounded $p$-Laplacian, optimal regularity conjecture. The optimal regularity…

Let $\Omega\subset\mathbb{R}^N$ be the domain under consideration, let $p>1$, and let $\{u_n\}\subset C^1(\overline{\Omega})$ satisfy, in the weak sense, for some positive constant…

Let $B$ be a ball and let $d4b\Omega\subset\mathbb{R}^N$ be a domain. Denote by $d706_{1,p}(\Omega)$ the first eigenvalue of the Dirichlet $p$-Laplacian on $d4b\Omega$, and by…

Let $\mathcal{R}_a$ denote the rectangle considered in the paper, let $\lambda_2(p;\mathcal{R}_a)$ be its second eigenvalue for the pure $p$-Laplacian, and let…

Variable-exponent conjecture. Theorems concerning multiplicity for … should remain valid when $q$ is replaced by a variable exponent $q$.

Let $M$ be a metric-measure space, let $up1$, and let $Omegasubset M$ be a connected open set containing a fixed origin $o$. Write $r(x)=operatorname{dist}(x,o)$, and let…

BCN Conjecture. If $\Omega$ is a smooth domain with connected complement $\Omega^c$, then $\Omega$ is either a ball, a half-space, a generalized cylinder…

Let $\Omega=(0,1)^2$ be the unit square. Let ${\cal S}_1$ be the class of functions odd about one of the square's middle lines and even about the other, and let ${\cal S}_2$ be the…

Let $p>1$, and let $\lambda_{1,p}(\Omega)$ denote the first eigenvalue of the $p$-Laplacian on a bounded domain $\Omega\subset\mathbb{R}^2$, with $|\Omega|$ denoting its area. Isop…

Optimal-threshold conjecture. For $p>1$, this threshold should be optimal for obtaining solutions in $W_0^{1,p}(\Omega)$ for positive data $f\in L^m(\Omega)$, $m>1$.

Complete classification conjecture. Either $u\equiv 0$, or there exist $\lambda:=u(0)^{\frac{p}{N-p}}>0$ and $x_{0}\in\mathbb{R}^{N}$ such that

Let $\Omega$ be a domain, let $u$ solve the $p$-Laplacian equation … and let $r$ be the local integrability exponent. The vector field $|\nabla u|^{p-2}\nabla u$ is the stress fiel…

Let $d\geq 2$, let $\nabla_A$ denote the magnetic gradient, let $h_{A,p}$ be the corresponding magnetic $p$-Laplacian form domain, and let $\mu_{p,d}$ be the Hardy constant. Let…

Robin maximization conjecture. The ball maximizes $\lambda(\Omega,2,\beta)$ among all smooth bounded domains $\Omega$ of fixed volume.

The $C^{p'}$ regularity conjecture. A bounded solution should satisfy

Let $B\subset\mathbb{R}^2$ be a disk and let $P\subset B$ be an inner domain with dihedral symmetry of order $n$, where $n\geq 2$ is odd. Consider $\Omega=B\setminus\overline{P}$ a…

Let $B_r$ and $B_s$ be open disks in $\mathbb{R}^2$ of radii $r$ and $s$, respectively, with $B_s$ contained in $B_r$. Let $\lambda_1(B_r\setminus\overline{B_s})$ denote the first…

Let $N\geq 3$, $1\leq p<N$, and let $p^{\ast}$ denote the critical Sobolev exponent. For a function $u$ on $\mathbb{Z}^{N}$, define the discrete $p$-Laplacian by … for $p>1$, with…

Let $\Omega\subset\mathbb R^N$ be the domain, let $p>1$, let $\lambda>0$, and let $\sigma(-\Delta_p)$ denote the spectrum of the Dirichlet $p$-Laplacian. Theorems 1 and 2 assert gr…

Extension conjecture. With the exception of Lemma …

Parameter-scaling conjecture. The relation

Regularity and minimal-growth solution conjecture. Then: