42 problems

Let $p>1$ and let $\omega_p$ denote the optimal discrete $p$-Hardy weight. A function is absolutely monotonic on an interval if all of its derivatives are non-negative there. Absol…

Badiale–Tarantello conjecture. The best constant in this inequality is

Let $s\in(0,1)$, and consider the fractional Laplacian associated with a censored stable process on a convex domain. The best Hardy constant is the largest constant for which the c…

Let $\Omega\subset\mathbb{R}^n$, with $n\geq 2$, be a convex domain, and let $C(\Omega)$ denote a positive constant for which … where…

Let $U$ be a bounded region in ${\bf R}^{N}$, let $H=-\Delta_{\mathrm{DIR}}$ act in $L^{2}(U)$, and let $d(x)$ be the distance from $x$ to a closed subset of…

Let $U$ be a region, let $H$ be a uniformly elliptic second-order operator on $L^{2}(U)$ with Dirichlet boundary conditions, and let $Q$ be its associated quadratic form. For…

Wannebo's conjecture. The decompositions

Odd-order decomposition conjecture. The decomposition above holds for all odd $m$, $p>1$, $N$, and open $\Omega\subset\mathbb{R}^N$; for even $m$, there are $p$, $N$, and $\Omega$…

Full-parameter properties conjecture. The following assertions hold:

Positive-coefficient expansion conjecture. For every $n\geq\ell$, this is a convergent series expansion and $A_k^{(\ell,p)}>0$ for every $k\in\mathbb{N}$.

Let $N>p$ and let $\mathbb{R}^N_+=\{x\in\mathbb{R}^N:x_N>0\}$. For $p^*:=Np/(N-p)$, let $u\in C_0^\infty(\mathbb{R}^N)$ and let $\gamma>p-1$. Weighted Hardy-Sobolev-Maz'ya conjectu…

Let $b_*$ be the critical parameter associated with the sharp Hardy-type inequality in the cylindrical-spherical weighted setting, and let $d$ and $k$ denote the ambient and cylind…

Non-attainment conjecture. The solution satisfies

Davies's conjecture. For every bounded domain $\Omega$ and every $y\in\partial\Omega$,

Let $\iota_3:(0,1)\to\mathbb{R}$ be the function defined by … Here $t_2^*$ is the endpoint appearing in the reduced five-bubble analysis. Second critical point conjecture. There sh…

Let $f_5$ be the limit function whose critical point determines the parameters $(\lambda_0,\lambda_1,\lambda_2,t)$ in the five-bubble existence theorem, and let $u_\varepsilon$ den…

Fischer–Keller–Pogorzelski positivity conjecture. All coefficients $c_l$ in this expansion are strictly positive for every $p>1$. The conjecture asserts positivity beyond the integ…

Conjecture. The constant $mu_{n+1,\beta}(\Omega)$ is attained in $W_0^{1,2}(\Omega)$ only if

Aharonov–Bohm Hardy conjecture. If $\beta\notin\mathbb{Z}$, then there exists a constant $\lambda(p)>\left(\frac{2-p}{p}\right)^p$ such that

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…

Let $L=-\Delta+|x|^2$ be the Hermite operator on $\mathbb{R}$, and let $h_k$, $k=0,1,2,\ldots$, be the normalized one-dimensional Hermite functions. For $f\in H^1(\mathbb{R})$, wri…

Sign conjecture for $b_k(\alpha)$. For every $\alpha \in (0,1/3) \cup [5, \infty)$, there exists an integer $i \geq 1$ such that $b_i(\alpha)<0$; equivalently, the constants…

Let $N\geq 5$ and let $u\in C_c^{\infty}(\mathbb{H}^{N}\setminus\{x_0\})$. In the case $\lambda=0$, the second-order Heisenberg–Pauli–Weyl inequality is … Here $r$ is the hyperboli…

Let $N\geq 5$, let $0\leq\lambda\leq\lambda_1(\mathbb{H}^{N})$, and let $u\in C_c^{\infty}(\mathbb{H}^{N}\setminus\{x_0\})$. The second-order Heisenberg–Pauli–Weyl inequality is ……

Let $B_R$ be the ball of radius $R$ in $\mathbb{R}^N$, let $N>k\geq 2$, and let $u\in C_c^k(B_R)$. Critical logarithmic Hardy conjecture. The inequality … holds. Moreover, the cons…