8 problems

Let $(\Omega,h)$ be a quantum disc as defined in the cited work, and let $\mathbf{K}(t)$ denote the corresponding heat-content quantity. Quantum-disc heat-content conjecture. As…

Consider the heat content functional for an $N$-gon under a fixed-area constraint, with $t\in\mathbb R$, and let the regular $N$-gon be the symmetric critical configuration. Local…

Let $\Omega\subset \mathbb{R}^d$ be an open bounded set with smooth boundary, and let … be its heat content with respect to the Poisson kernel. Acuna's heat-content perimeter conje…

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, and let $p_t(x)$ denote its heat-content quantity. Heat-content inradius conjecture. If there is a constant $c_1>0$ such that…

Let $(M,g)$ be a compact smooth manifold without boundary. Let $u$ be a Laplacian eigenfunction with eigenvalue $\lambda$, and let $p_t(x)$ be globally defined with respect to its…

Let $(M,g)$ be a compact smooth manifold without boundary. Let $p_t(x)$ denote the heat-content quantity used in the paper, and let $\mathcal{H}^{n-1}(\partial N)$ be the $(n-1)$-d…

Let $M$ be a compact Riemannian manifold with smooth boundary $\partial M$, let $\delta$ denote the distance to the boundary, let $D$ be an operator of Laplace type with Dirichlet…

Let $M$ be a compact Riemannian manifold with smooth boundary $partial M$, let $delta$ denote the geodesic distance to the boundary, let $D$ be an operator of Laplace type with Dir…