8 problems

Let $A$ be a cone differential operator on the manifold with conical singularity ${\mathbb B}$, let $\mathcal{D}(A_{\min})$ denote the domain of its minimal extension, and let…

Let $d\in\mathbb{N}$, let $p$ and $\gamma$ be parameters for which the weighted Sobolev spaces below are defined, let $X$ be the underlying Banach space, and let…

Compactness-rate conjecture. There exist $C>0$ and $N\in\mathbb{N}$ such that, for every $m\geq N$ and every $u\in\overline{\operatorname{Span}\{q_j\}_{j>m}}^{H^1(\nu)}$,

Let $V(x)=x^{2k}/(2k)+O(x^{2k})$ be an even polynomial potential of degree $2k$. Let $(b_n)_{n\in\mathbb{N}}=(a_n^2)_{n\in\mathbb{N}}$ be the positive sequence defined by the recur…

Non-attainment conjecture. The solution satisfies

Let $p$ and $\hat p$ be the integrability exponents appearing in Theorem 3.4(c), with $p\in(1,\hat p]$. Theorem 3.4(c) extension conjecture. The result of Theorem 3.4(c) should hol…

Let $h_0$ be the initial perturbation, and let $\eta(h_0)$ be the quantity used in the estimates to control the perturbation. The notation $H^2_{\lambda+2}$ denotes the weighted So…

Exponential bound conjecture. The proof gives