Related papers: Heat content asymptotics of some random Koch type …

In this short paper, we derive an alternative proof for some known [van den Berg & Gilkey 2015] short-time asymptotics of the heat content in compact full-dimensional submanifolds $S$ with smooth boundary. This includes formulae like…

We consider a heat problem with discontinuous diffusion coefficientsand discontinuous transmission boundary conditions with a resistancecoefficient. For all compact $(\epsilon,\delta)$-domains $\Omega\subset\mathbb{R}^n$ with a $d$-set…

This paper studies the small time behavior of the heat content of rotationally invariant $\alpha$--stable processes, $0<\alpha \leq 2$, in domains in $\R^d$. Unlike the asymptotics for the heat trace, the behavior of the heat content…

We propose a novel approach for studying small-time asymptotics of the fractional heat content of $C^2$ non-characteristic domains in Carnot groups. Denoting the sub-Laplacian operator by $\mathcal{L}$, the fractional heat content of a…

This paper studies the heat content} for Schr\"odinger operators of the fractional Laplacian $(-\Delta)^{\alpha/2}$, $0<\alpha\leq 2$ in $R^d$, $d\geq 1$. Employing probabilistic and analytic techniques, a small time asymptotic expansion…

The spectral heat content of a domain $\Omega\subset\mathbb{R}^d$ corresponding to a $d$-dimensional stochastic process $X=(X_t)_{t\ge 0}$ is defined as \[Q^{X}_\Omega(t)=\int_{\mathbb{R}^d} \mathbb{P}_x(\tau^X_\Omega>t)dx,\] where…

In this paper we study the small time asymptotic behavior of the spectral heat content $\widetilde{Q}_D^{(\alpha)}(t)$ of an arbitrary bounded $C^{1,1}$ domain $D$ with respect to the \textit{subordinate killed Brownian motion} in $D$ via…

The aim of our paper is twofold. First, we present new mathematical developments on the analysis of de Gennes' hypothesis on the short-time asymptotics of the heat content for bounded domains with smooth boundary and with fractal boundary.…

We study the small-time asymptotics of the heat content of smooth non-characteristic domains of a general rank-varying sub-Riemannian structure, equipped with an arbitrary smooth measure. By adapting to the sub-Riemannian case a technique…

We study the heat content asymptotics with either Dirichlet or Robin boundary conditions where the initial temperature exhibits radial blowup near the boundary. We show that there is a complete small-time asymptotic expansion and give…

We study the spectral heat content for a class of open sets with fractal boundaries determined by similitudes in $\mathbb{R}^{d}$, $d\geq 1$, with respect to subordinate killed Brownian motions via $\alpha/2$-stable subordinators and…

We consider weak solutions of the fractional heat equation posed in the whole $n$-dimensional space, and establish their asymptotic convergence to the fundamental solution as $t\to\infty$ under the assumption that the initial datum is an…

The relative heat content associated with a subset $\Omega\subset M$ of a sub-Riemannian manifold, is defined as the total amount of heat contained in $\Omega$ at time $t$, with uniform initial condition on $\Omega$, allowing the heat to…

This paper establishes the small-time asymptotic behaviors of the regular heat content and spectral heat content for general Gaussian processes in both one-dimensional and multi-dimensional settings, where the boundary of the underlying…

Let $(M^n, g)$ be a complete Riemannian manifold with $Rc\geq -Kg$, $H(x, y, t)$ is the heat kernel on $M^n$, and $H= (4\pi t)^{-\frac{n}{2}}e^{-f}$. Nash entropy is defined as $N(H, t)= \int_{M^n} (fH) d\mu(x)- \frac{n}{2}$. We studied the…

In this paper we study the heat content for sets with positive reach. In details, we investigate the asymptotic behavior of the heat content of bounded subsets of the Euclidean space with positive reach. The concept of positive reach was…

We obtain monotonicity and convexity results for the heat content of domains in Riemannian manifolds and in Euclidean space subject to various initial temperature conditions. We introduce the notion of a strictly decreasing temperature set,…

We identify the short time asymptotics of the sub-Riemannian heat content for a smoothly bounded domain in the first Heisenberg group. Our asymptotic formula generalizes prior work by van den Berg-Le Gall and van den Berg-Gilkey to the…

Let $D$ be a bounded, connected, open set in Euclidean space $\mathbb{R}^{2}$ with polygonal boundary. Suppose $D$ has initial temperature $1$ and the complement of $D$ has initial temperature $0$. We obtain the asymptotic behaviour of the…

It is shown that certain kinds of behavior, which hitherto were expected to be characteristic for classical gravity and quantum field theory in curved spacetime, as the infinite dimensional Bondi-Metzner-Sachs symmetry, holography on event…