18 problems

Let $M^n$ be a complete Riemannian manifold, let $\Omega\subset M^n$ be a bounded domain, and let $\lambda_k$ denote the Dirichlet eigenvalues of the operator $\mathcal{L}$ from Pr…

Let $P u=-\Delta u+V(x)u+u$ be subcritical in $\mathbb{R}^n$, and assume that … If $u$ is a positive solution of $Pu=0$ in $\mathbb{R}^n$ satisfying the stated condition, then let…

Let $n\ge 3$, let $\mathcal{E}$ be the class of all complex elliptic operators under study, and for $L\in\mathcal{E}$ let $p_-(L)$ be the lower endpoint of the maximal interval of…

Let $M$ be a hyperkähler manifold that is AC of order $l\geq 1$, and let $k\geq -1$ be an integer. Let $x:M\rightarrow\mathbb{H}$ be a smooth function such that $D(x)$ is a $1$-for…

Generic gap-edge conjecture. Each endpoint of the gap is a unique, modulo the dual lattice, nondegenerate extremum of a single band function $\lambda_j(k)$.

Let $u$ be a super-exponentially decaying solution of a second-order periodic, or more generally an appropriate class of periodic elliptic, equation. Landis-type conjecture. Such a…

Let $L$ be a periodic second-order elliptic operator with sufficiently nice coefficients. Absolute continuity conjecture. The spectrum of $L$ is absolutely continuous, and hence…

BMS conjecture. Assume that $(M,g)$ is geodesically complete. Then $(M,g)$ is $L^{2}$-positivity preserving.

Let $E_{b4}(Z)$ be the operator associated with the elliptic complex on the periodic end, and let $\widehat{H}^*_z(V)$ denote the corresponding periodic cohomology group, with para…

Let $M$, $\beta$, $\theta$, $\rho$, $u$, $b$, and the conditions defining the range … be as in Theorem … is replaced by … , but the supplied text gives no resolution or further co…

Strong ratio-limit conjecture. Locally uniformly for $(x,y)\in M\times M$,

Let $\mathfrak{A}: [0,T] \to \mathcal{L}(V,V^*)$ be the operator associated with a non-autonomous form $\mathfrak{a}$, and suppose that … The one-half-Hölder maximal regularity cou…

Ground-state domination conjecture. Then $\lambda$ belongs to the spectrum of the Friedrichs extension of $P$ on $L^2(\Omega,\mathrm{d}x)$. This conjecture links pointwise dominati…

Let $M$ be a Riemannian manifold, and let $P_1$ and $P_2$ be subcritical elliptic operators of the form specified in the paper. Two heat kernels are equivalent when they are compar…

Let $M$ be a connected noncompact Riemannian manifold, and let $k_P^M(x,y,t)$ denote the positive minimal heat kernel of a second-order elliptic operator $P$ on $M$. An operator is…

Let $X$ be a p.c.f. fractal, let $P$ be a polynomial, and let $P(\Delta)$ denote the associated operator on the product space $X^m$. The operator $P(\Delta)$ is elliptic when…

Vanishing of the inner-index conjecture. The inner index vanishes:

Sectorial-projection criterion. There is a criterion for weak unique continuation for $A$ in terms of the range of $P_+(B_0)$.